Heat transfer enhancement and entropy generation analysis of Al₂O₃-water nanofluid in an alternating oval cross-section tube using two-phase mixture model under turbulent flow

Abstract

Heat transfer and turbulent flow of Al₂O₃-water nanofluid within alternating oval cross-section tube are numerically simulated using Eulerian-Eulerian two-phase mixture model. The primary goal of the present study is to investigate the effects of nanoparticles volume fraction, nanoparticles diameter and different inlet velocities on heat transfer, pressure drop and entropy generation characteristics of the alternating oval cross-section tube. For numerical simulation validation, the numerical results were compared with experimental data. Also, constant wall temperature boundary condition was considered on the tube wall. In addition, the comparison of thermal-hydraulic performance and the entropy generation characteristics between alternating oval cross-section tube and circular tube under same fluids were done. The results show that the heat transfer coefficient and pressure drop of alternating oval cross-section tube is more than base tube under same fluids. Also, these two parameters are increased when adding Al₂O₃ nanoparticle into water fluid, at any inlet velocity for both tubes. Furthermore, compared to the base fluid, the value of the heat transfer enhancement of nanofluid is higher than the increase of friction factor of nanofluid at the same given inlet boundary conditions. The results of entropy generation analysis illustrate that the total entropy generation increase with increasing the nanoparticles volume fraction and decreasing the nanoparticles diameter of nanofluid. The generation of thermal entropy is the main part of irreversibility, and Bejan number with an increase of the nanoparticles diameter slightly increases. Finally, at any given inlet velocity the frictional irreversibility is grown with an increase the nanoparticles volume fraction.

Appendix

Let the reference parameters are shown by the subscript 0. Here we define the non-dimensional parameters as [28, 29]:

$$ {\displaystyle \begin{array}{c}{V}_m^{\ast }=\frac{V_m}{V_0},{\rho_{nf}}^{\ast }=\frac{\rho_{nf^{\ast }}}{\rho_0},{\rho_1}^{\ast }=\frac{\rho_1}{\rho_{1,0}},{\rho_2}^{\ast }=\frac{\rho_2}{\rho_{2,0}},{\nabla}^{\ast }={L}_0\nabla \\ {}{P}^{\ast }=\frac{P}{\rho_0{V}_0^2},{V^{\ast}}_{dr,2}=\frac{V_{dr,2}}{V_{0\; dr,2}},{T}^{\ast }=\frac{T-{T}_0}{T_w-{T}_0}\end{array}} $$

(33)

Substituting for dimensionless parameters gives,

Non-dimensional mixture continuity equation:

$$ {\nabla}^{\ast}\bullet \left({\uprho_{\mathrm{nf}}}^{\ast }\ {{\mathrm{V}}_{\mathrm{m}}}^{\ast}\right)=0 $$

(34)

Non-dimensional momentum equation for mixture:

$$ {\nabla}^{\ast}\bullet \left({\uprho_{\mathrm{nf}}}^{\ast }\ {{\mathrm{V}}_{\mathrm{m}}}^{\ast}\kern0.50em {{\mathrm{V}}_{\mathrm{m}}}^{\ast}\right)=-{\nabla}^{\ast }{\mathrm{P}}^{\ast }+\frac{1}{{\mathrm{N}}_{\mathrm{Re}}}{\nabla}^{\ast}\bullet {\left(\uptau +{\uptau}_{\mathrm{t}}\right)}^{\ast }-{\mathrm{N}}_{\uprho}{{\mathrm{N}}_{\mathrm{D}}}^2{\nabla}^{\ast}\bullet \left(\frac{\upvarphi_2}{1-{\upvarphi}_2}\frac{\uprho_1^{\ast }\ {\uprho}_2^{\ast }}{{\uprho_{\mathrm{nf}}}^{\ast }}\ {{\mathrm{V}}^{\ast}}_{\mathrm{dr},2}\ {{\mathrm{V}}^{\ast}}_{\mathrm{dr},2}\right) $$

(35)

Non-dimensional stress equation for mixture:

$$ {\left(\uptau +{\uptau}_{\mathrm{t}}\right)}^{\ast }=\frac{\left(\uptau +{\uptau}_{\mathrm{t}}\right)}{{\mathrm{V}}_0\left({\upmu}_{\mathrm{t},\mathrm{m}}+{\upmu}_{\mathrm{nf}}\right)/{\mathrm{L}}_0} $$

(36)

Non-dimensional volume fraction equation:

$$ {\nabla}^{\ast}\bullet \left(\ {\upvarphi}_2\ {\uprho}_2^{\ast }{{\mathrm{V}}_{\mathrm{m}}}^{\ast}\right)=-{\mathrm{N}}_{\mathrm{D}}\ {\nabla}^{\ast}\bullet \left({\upvarphi}_2\frac{\uprho_2^{\ast }\ {\uprho}_2^{\ast }}{{\uprho_{\mathrm{nf}}}^{\ast }}\ {\mathrm{V}}_{\mathrm{dr},2}\right) $$

(37)

where,

$$ {\mathrm{N}}_{\mathrm{Re}}=\frac{\uprho_0{\mathrm{V}}_0\ {\mathrm{L}}_0}{\upmu_0}\kern0.5em \left(\mathrm{Reynolds}\ \mathrm{Number}\right) $$

(38)

$$ {N}_{\rho }=\frac{\rho_{2,0}}{\rho_{1,0}}\kern0.5em \left(\mathrm{Density}\ \mathrm{Ratio}\right) $$

(39)

$$ {\mathrm{N}}_{\mathrm{D}}=\frac{\uprho_{1,0}{\mathrm{V}}_{0\ \mathrm{dr},2}}{\uprho_0{\mathrm{V}}_0}\kern0.5em \left(\mathrm{Drift}\ \mathrm{Number}\right) $$

(40)

Non-dimensional energy equation:

$$ {\nabla}^{\ast}\bullet \left(\sum \limits_{\mathrm{k}=1}^2{\upvarphi}_{\mathrm{k}}\ {\uprho}_{\mathrm{k}}^{\ast }\ {\mathrm{C}}_{\mathrm{pk}}^{\ast }\ {\mathrm{V}}_{\mathrm{k}}^{\ast }\ {\mathrm{T}}^{\ast}\right)=\frac{1}{{\mathrm{N}}_{\mathrm{Pe}}}{\nabla}^{\ast}\bullet \left({\mathrm{K}}_{\mathrm{eff}}^{\ast }\ {\nabla}^{\ast }{\mathrm{T}}^{\ast}\right) $$

(41)

where,

$$ {\mathrm{N}}_{\mathrm{P}\mathrm{e}}=\frac{\uprho_0{\mathrm{C}}_{\mathrm{P}0}{\mathrm{V}}_0\ {\mathrm{L}}_0}{{\mathrm{K}}_0}\kern0.5em \left(\mathrm{Peclet}\ \mathrm{number}\right) $$

(42)

$$ {\mathrm{C}}_{\mathrm{pk}}^{\ast }=\frac{{\mathrm{C}}_{\mathrm{pk}}}{{\mathrm{C}}_{\mathrm{P}0}} $$

(43)

$$ {\mathrm{K}}_{\mathrm{eff}}^{\ast }=\frac{{\mathrm{K}}_{\mathrm{eff}}}{{\mathrm{K}}_0} $$

(44)

Non-dimensional turbulent kinetic energy and dissipation:

$$ {\nabla}^{\ast}\bullet \left({\uprho_{\mathrm{nf}}}^{\ast}\kern0.50em {{\mathrm{V}}_{\mathrm{m}}}^{\ast }\ {\mathrm{k}}^{\ast}\right)=\frac{1}{{\mathrm{N}}_{\mathrm{Re}}}{\nabla}^{\ast}\bullet \left[\left({\upmu_{\mathrm{nf}}}^{\ast }+\frac{{\upmu_{\mathrm{t},\mathrm{m}}}^{\ast }}{\upsigma_{\mathrm{k}}}\right)\ {\nabla}^{\ast }{\mathrm{k}}^{\ast}\right]+{{\mathrm{G}}_{\mathrm{k},\mathrm{m}}}^{\ast }-\frac{1}{{\mathrm{N}}_{\mathrm{Re}}}{\uprho_{\mathrm{nf}}}^{\ast }\ {\upvarepsilon}^{\ast } $$

(45)

$$ {\nabla}^{\ast}\bullet \left({\uprho_{\mathrm{nf}}}^{\ast}\kern0.50em {{\mathrm{V}}_{\mathrm{m}}}^{\ast }\ {\upvarepsilon}^{\ast}\right)=\frac{1}{{\mathrm{N}}_{\mathrm{Re}}}{\nabla}^{\ast}\bullet \left[\left({\upmu_{\mathrm{nf}}}^{\ast }+\frac{{\upmu_{\mathrm{t},\mathrm{m}}}^{\ast }}{\upsigma_{\upvarepsilon}}\right)\ {\nabla}^{\ast }{\varepsilon}^{\ast}\right]+\frac{1}{{\mathrm{N}}_{\mathrm{Re}}}\frac{\varepsilon^{\ast }}{{\mathrm{k}}^{\ast }}\ \left({\mathrm{C}}_{1\upvarepsilon}\ {{\mathrm{G}}_{\mathrm{k},\mathrm{m}}}^{\ast }-{\mathrm{C}}_{2\upvarepsilon}{\uprho_{\mathrm{nf}}}^{\ast }\ {\upvarepsilon}^{\ast}\kern0.5em \right) $$

(46)