Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 1, pp 195–206 | Cite as

Presentation of new thermal conductivity expression for \(\hbox {Al}_2\hbox {O}_3\)–water and \(\hbox {CuO}\)–water nanofluids using gene expression programming (GEP)

  • Saber Yekani Motlagh
  • Abbas SharifiEmail author
  • Mohsen Ahmadi
  • Homayoun Badfar


In the present investigation, gene expression programming (GEP) is used to predict thermal conductivity of nanofluids consisting of \(\hbox {Al}_2\hbox {O}_3\) and CuO nanoparticles suspended in water. The obtained new model is a function of temperature, volume fraction, and diameter of the nanoparticles. To predict the thermal conductivity, experimental data from literatures were partitioned into two sets: a train (800 numbers) and a test (200 numbers) data sets. The model was designed by the train set, and the results were compared with the test data set. The estimated heat conductivity was compared with experimental data and several most cited relations in the literature. Moreover, the new thermal conductivity model was tested in the simulation of benchmark case study of nanofluid free convection inside the square cavity. The predicted Nusselt number was compared with available experimental data at five Rayleigh numbers. The findings illustrated that the GEP can estimate and model the thermal conductivity of nanofluid very efficiently, and it can be used successfully for simulation of engineering problems.

Graphical Abstract


Nanofluid Thermal conductivity Genetic expression programming Modeling 



The Akaike information criterion


Artificial neural network


Correlation coefficient


Gene expression programming


Mean square error \(\sum _{j=1}^{n}(C_{\mathrm{j}}-T_\mathrm{j})^2/n\)


Root-mean-square error \(\sqrt{\hbox {MSE}}\)


Random numerical constant


Coefficient of determination \(\sum _{j=1}^{n}(C_{\mathrm{j}}-T_{\mathrm{avg}})/\sum _{j=1}^{n}(T_{\mathrm{j}}-T_{\mathrm{avg}})\)


Sum of squared error \(\sum _{j=1}^{n}(C_{\mathrm{j}}-T_\mathrm{j})^2\)


a, b, c

Terminals as temperature, volume fraction, and diameter, respectively


The value predicted by chromosome i for fitness case j


Fitness function


The jth constant of ith gene


Natural logarithm, cosine, power, logarithm, square root, exponential functions


The target value


Random numerical constants

Fluid mechanic


Specific heat at constant pressure


Particle diameter




Characteristic length


Mean free path


Thermal conductivity


Boltzmann constant \(=1.38064852\times 10^{-23}\) m\(^2\) kg s−2 K−1


Nusselt number (hL)/k


Prandtl number \((C_{\mathrm{p}}\mu )/ k\)


Rayleigh number \((g \beta Pr \Delta T L^2)/\nu ^2\)


Brownian Reynolds number \(\frac{\rho _{\mathrm{bf}}k_\mathrm{B} T}{3 \pi \mu ^2 l_{\mathrm{bf}}}\)


Fluid temperature


Fluid velocity


Ave, \(h,c,*,r\)

Average, hot, cold, non-dimensional, relative


Fluid, nanofluid, base fluid, particle

Greece symbol


Thermal diffusion coefficient


Thermal expansion coefficient




Volume fraction

\(\sigma ^2\)

Residual variance


Kinematic viscosity


Dynamic viscosity


Fluid density


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUrmia University of Technology (UUT)UrmiaIran
  2. 2.Department of Industrial EngineeringUrmia University of Technology (UUT)UrmiaIran

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