# Taguchi optimization for natural convection heat transfer of Al_{2}O_{3} nanofluid in a partially heated cavity using LBM

- 24 Downloads

## Abstract

In the present study for the first time, Taguchi approach was applied to specify the optimal condition of the parameters in the natural convection heat transfer of Al_{2}O_{3} nanofluid for a partially heated cavity. The flow and energy equations are solved by the lattice Boltzmann method. The influence of the 5 factors including Rayleigh number, position, hot length, cold length, volume concentration of the Al_{2}O_{3} nanoparticles is examined. The Nusselt number on the hot section is measured for the response factor. In Taguchi optimization method, the levels of every factor were fixed at 3 levels and the L27 orthogonal array. The conclusions of the Taguchi–LBM technique indicated that the optimum conditions were attained at the maximum Rayleigh number, cold length and volume fraction and the minimum hot length in the bottom–bottom configuration in the variety of the design parameters. Also, the most significant parameter influencing the Nusselt number on the hot wall was the Rayleigh number, while changing the volume fraction had a negligible effect.

## Keywords

Partially heated Natural convection Nanofluid Lattice Boltzmann Taguchi optimization## List of symbols

- \(C\)
Lattice speed

- \(C_{\text{p}}\)
Specific heat

- dB
Decibel

*DF*Degrees of freedom

- \(\vec{e}_{\text{i}}\)
Velocity in discrete direction \(i\)

- \(F\)
*F*value- \(f_{\text{i}}\)
Particle distribution function for velocity field

- \(g_{\text{i}}\)
Particle distribution function for thermal field

- \(g_{\text{y}}\)
Gravitational acceleration in the \(y\) direction

- \(H\)
Height and width of enclosure

- \(\dot{B}_{\text{i}}\)
Buoyant body force term

- \(k_{\text{t}}\)
Thermal conductivity

*L*Length

- \(M\)
Total number of discrete lattice directions

- \(Ma\)
Mach number

- \({\text{MS}}\)
Mean squares

- \(n\)
Number of case iteration

- \(\vec{n}\)
Outer normal unit vector

- \(Nu\)
Nusselt number

- \(P\)
Position

- \(Pr\)
Prandtl number

- \(R\)
Ideal gas constant

- \(Ra\)
Rayleigh number

- \(\vec{r}\)
Position vector

- \(s\)
Geometric distance

- \({\text{SS}}\)
Sums of squares

- SNR
Signal-to-noise ratio

*T*Temperature

- \(\vec{u}\)
Macroscopic velocity vector

- \(x,y\)
\(x\)- and \(y\)-coordinate system

*t*Time

- \(X,Y\)
Dimensionless coordinate of the 2D rectangular cavity

- \(y_{\text{n}}\)
Measured response

## Greek symbols

- \(\alpha\)
Thermal diffusion

- \(\beta\)
Coefficient of thermal expansion

- \(\eta\)
Value of predicted SNR

- \(\theta\)
Dimensionless temperature

- \(\nu\)
Kinematic viscosity

- \(\rho\)
Density

- \(\tau_{\text{t}}\)
Relaxation time for thermal field

- \(\tau_{\upnu}\)
Relaxation time for velocity field

- \(\phi\)
Volume fraction

## Subscripts

- \({\text{nf}}\)
Nanofluid

- \(C,H\)
Cold, hot

- \(i\)
Index for the discrete direction

*T*Total

*w*Wall

*p*Particle

## Superscripts

- (eq)
Equilibrium

## Notes

### Acknowledgements

The authors would like to gratefully acknowledge the Ferdowsi University of Mashhad, Mashhad, Iran, for their support and funding (No. 47970) provided for the research.

## References

- 1.Choi S, Estman J. Enhancing thermal conductivity of fluids with nanoparticles. ASME-Publications-FED. 1995;231:99–106.Google Scholar
- 2.Sobhani M, Behzadmehr A. Investigation of thermo-fluid behavior of mixed convection heat transfer of different dimples-protrusions wall patterns to heat transfer enhancement. Heat Mass Transf. 2018;54(11):3219–29.CrossRefGoogle Scholar
- 3.Gahrooei HRE, Ghazanfari MH. Application of a water based nanofluid for wettability alteration of sandstone reservoir rocks to preferentially gas wetting condition. J Mol Liq. 2017;232:351–60.CrossRefGoogle Scholar
- 4.Vasilache V, Popa C, Filote C, Cretu MA, Benta M. Nanoparticles applications for improving the food safety and food processing. In: 7th international conference on materials science and engineering—BRAMAT Braşov, February 2011.Google Scholar
- 5.Kabeel A, Omara Z, Essa F. Numerical investigation of modified solar still using nanofluids and external condenser. J. Taiwan Inst Chem Eng. 2017;75:77–86.CrossRefGoogle Scholar
- 6.Keyhani M, Prasad V, Cox R. An experimental study of natural convection in a vertical cavity with discrete heat sources. ASME J Heat Transf. 1988;110:616–24.CrossRefGoogle Scholar
- 7.Valencia A, Frederick RL. Heat transfer in square cavities with partially active vertical walls. Int J Heat Mass Transf. 1989;32(8):1567–74.CrossRefGoogle Scholar
- 8.da Silva AK, Lorente S, Bejan A. Optimal distribution of discrete heat sources on a wall with natural convection. Int J Heat Mass Transf. 2004;47:203–14.CrossRefGoogle Scholar
- 9.Mahapatra PS, Manna NK, Ghosh K. Effect of active wall location in a partially heated enclosure. Int Commun Heat Mass Transf. 2015;61:69–77.CrossRefGoogle Scholar
- 10.Oztop HF, Abu-Nada E. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int J Heat Fluid Flow. 2008;29(5):1326–36.CrossRefGoogle Scholar
- 11.Aminossadati SM, Ghasemi B. Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure. Eur J Mech B Fluids. 2009;28(5):630–40.CrossRefGoogle Scholar
- 12.Aminossadati SM, Ghasemi B. Natural convection of water–CuO nanofluid in a cavity with two pairs of heat source-sink. Int Commun Heat Mass Transf. 2011;38(5):672–8.CrossRefGoogle Scholar
- 13.Sheikhzadeh GA, Arefmanesh A, Kheirkhah MH, Abdollahi R. Natural convection of Cu–water nanofluid in a cavity with partially active side walls. Eur J Mech B Fluids. 2011;30(2):166–76.CrossRefGoogle Scholar
- 14.Jmai R, Ben-Beya B, Lili T. Heat transfer and fluid flow of nanofluid-filled enclosure with two partially heated side walls and different nanoparticles. Superlattices Microstruct. 2013;53:130–54.CrossRefGoogle Scholar
- 15.Mohebbi R, Rashidi MM. Numerical simulation of natural convection heat transfer of a nanofluid in an L-shaped enclosure with a heating obstacle. J Taiwan Inst Chem Eng. 2017;72:70–84.CrossRefGoogle Scholar
- 16.Mohebbi R, Izadi M, Chamkha AJ. Heat source location and natural convection in a C-shaped enclosure saturated by a nanofluid. Phys Fluids. 2017;29(12):122009.CrossRefGoogle Scholar
- 17.Krane RJ. Some detailed field measurements for a natural convection flow in a vertical square enclosure. In: Proceedings of the first ASME-JSME thermal engineering joint conference, 1983.
**1**: pp. 323–329.Google Scholar - 18.Ma Y, Mohebbi R, Rashidi MM, Yang Z. Simulation of nanofluid natural convection in a U-shaped cavity equipped by a heating obstacle: effect of cavity’s aspect ratio. J Taiwan Inst Chem Eng. 2018;93:263–76.CrossRefGoogle Scholar
- 19.Abu-Nada E, Oztop HF. Numerical analysis of Al
_{2}O_{3}/water nanofluids natural convection in a wavy walled cavity. Numer Heat Transf Part A Appl. 2011;59(5):403–19.CrossRefGoogle Scholar - 20.Izadi M, Mohebbi R, Karimiand D, Sheremet MA. Numerical simulation of natural convection heat transfer inside a ⊥ shaped cavity filled by a MWCNT-Fe3O4/water hybrid nanofluids using LBM. Chem Eng Process Process Intensif. 2018;125:56–66.CrossRefGoogle Scholar
- 21.Ranjbar P, Mohebbi R, Heidari H. Numerical investigation of nanofluids heat transfer in a channel consists of rectangular cavities by lattice Boltzmann method. Int J Mod Phys C. 2018;29(11):1–23.CrossRefGoogle Scholar
- 22.Matori A, Mohebbi R, Hashemi Z, Ma Y. Lattice Boltzmann study of multi-walled carbon nanotube (MWCNT)-Fe
_{3}O_{4}/water hybrid nanofluids natural convection heat transfer in a Π-shaped cavity equipped by hot obstacle. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7881-8.Google Scholar - 23.Cho CC. Heat transfer and entropy generation of natural convection in nanofluid-filled square cavity with partially-heated wavy surface. Int J Heat Mass Transf. 2014;77:818–27.CrossRefGoogle Scholar
- 24.Izadi M, Mohebbi R, Delouei AA, Sajjadi H. Natural convection of a magnetizable hybrid nanofluid inside a porous enclosure subjected to two variable magnetic fields. Int J Mech Sci. 2019;151:154–69.CrossRefGoogle Scholar
- 25.Nazari M, Kayhani MH, Mohebbi R. Heat transfer enhancement in a channel partially filled with a porous block: lattice Boltzmann method. Int J Mod Phys C. 2013;24(09):1350060.CrossRefGoogle Scholar
- 26.Saglam M, Sarper B, Aydin O. Natural convection in an enclosure with a discretely heated sidewall: heatlines and flow visualization. J Appl Fluid Mech. 2018;11(1):271–84.CrossRefGoogle Scholar
- 27.Ma Y, Mohebbi R, Rashidi MM, Yang Z, Sheremet MA. Numerical study of MHD nanofluid natural convection in a baffled U-shaped enclosure. Int J Heat Mass Transf. 2019;130:123–34.CrossRefGoogle Scholar
- 28.Cheikh NB, Ben Beya B, Lili T. Influence of thermal boundary conditions on natural convection in a square enclosure partially heated from below. Int Commun Heat Mass Transf. 2007;34(3):369–79.CrossRefGoogle Scholar
- 29.Abchouyeh MA, Mohebbi R, Fard OS. Lattice Boltzmann simulation of nanofluid natural convection heat transfer in a channel with a sinusoidal obstacle. Int J Mod Phys C. 2018;29(09):1850079.CrossRefGoogle Scholar
- 30.Succi S. The lattice Boltzmann method for fluid dynamics and beyond. Oxford: Oxford University Press; 2001. p. 0.8.Google Scholar
- 31.Mohamad AA. Lattice Boltzmann method: fundamentals and engineering applications with computer codes. Berlin: Springer; 2011.CrossRefGoogle Scholar
- 32.Lai FH, Yang YT. Lattice Boltzmann simulation of natural convection heat transfer of Al
_{0}O_{3}/water nanofluids in a square enclosure. Int J Therm Sci. 2011;50(10):1930–41.CrossRefGoogle Scholar - 33.Gangawane KM, Bharti RP, Kumar S. Effects of heating location and size on natural convection in partially heated Open-Ended enclosure by using Lattice Boltzmann method. Heat Transf Eng. 2016;37(6):507–22.CrossRefGoogle Scholar
- 34.Torabi M, Keyhani A, Peterson GP. A comprehensive investigation of natural convection inside a partially differentially heated cavity with a thin fin using two-set lattice Boltzmann distribution functions. Int J Heat Mass Transf. 2017;115:264–77.CrossRefGoogle Scholar
- 35.Ma Y, Mohebbi R, Rashidi MM, Manca O, Yang Z. Numerical investigation of MHD effects on nanofluid heat transfer in a baffled U-shaped enclosure using lattice Boltzmann method. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7518-y.Google Scholar
- 36.Sidik NAC, Razali SA. Lattice Boltzmann method for convective heat transfer of nanofluids—a review. Renew Sustain Energy Rev. 2014;38:864–75.CrossRefGoogle Scholar
- 37.Zhang DD, Wang L, Liu D, Zhao FY, Wang HQ. Free convective energy management of an inclined enclosure mounted with triple heating elements: multiple morphology optimizations with unique global energy supply. Int J Heat Mass Transf. 2017;115:406–20.CrossRefGoogle Scholar
- 38.Soleimani S, Ganji DD, Gorji M, Bararnia H, Ghasemi E. Optimal location of a pair heat source-sink in an enclosed square cavity with natural convection through PSO algorithm. Int Commun Heat Mass Transf. 2011;38(5):652–8.CrossRefGoogle Scholar
- 39.Kadiyala PK, Chattopadhyay H. Optimal location of three heat sources on the wall of a square cavity using genetic algorithms integrated with artificial neural networks. Int Commun Heat Mass Transf. 2011;38(5):620–4.CrossRefGoogle Scholar
- 40.Dias T Jr, Milanez LF. Optimal location of heat sources on a vertical wall with natural convection through genetic algorithms. Int J Heat Mass Transf. 2006;49(13–14):2090–6.CrossRefGoogle Scholar
- 41.Shirvan KM, Öztop HF, Al-Salem K. Mixed magnetohydrodynamic convection in a Cu-water-nanofluid-filled ventilated square cavity using the Taguchi method: a numerical investigation and optimization. Eur Phys J Plus. 2017;132(5):204.CrossRefGoogle Scholar
- 42.Katata-Seru L, Lebepe TC, Aremu OS, Bahadur I. Application of Taguchi method to optimize garlic essential oil nanoemulsions. J Mol Liq. 2017;244:279–84.CrossRefGoogle Scholar
- 43.Chou CS, Ho CY, Huang CI. The optimum conditions for comminution of magnetic particles driven by a rotating magnetic field using the Taguchi method. Adv Powder Technol. 2009;20(1):55–61.CrossRefGoogle Scholar
- 44.Gune S, Senyigit E, Karakaya E, Ozceyhan V. Optimization of heat transfer and pressure drop in a tube with loose-fit perforated twisted tapes by Taguchi method and grey relational analysis. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7824-4.Google Scholar
- 45.Abadeh A, Passandideh-Fard M, Maghrebi MJ, Mohammadi M. Stability and magnetization of Fe
_{3}O_{4}/water nanofluid preparation characteristics using Taguchi method. J Therm Anal Calorim. 2019;135(2):1323–34.CrossRefGoogle Scholar - 46.Wang XQ, Mujumdar AS. Heat transfer characteristics of nanofluids: a review. Int J Therm Sci. 2007;46(1):1–19.CrossRefGoogle Scholar
- 47.Brinkman H. The viscosity of concentrated suspensions and solutions. J Chem Phys. 1952;20(4):571.CrossRefGoogle Scholar
- 48.Sheikholeslami M, Ganji D. Nanofluid convective heat transfer using semi analytical and numerical approaches: a review. J Taiwan Inst Chem Eng. 2016;65:43–77.CrossRefGoogle Scholar
- 49.Dixit H, Babu V. Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method. Int J Heat Mass Transf. 2006;49(3):727–39.CrossRefGoogle Scholar
- 50.Sobhani M, Tighchi HA, Esfahani JA. Taguchi optimization of combined radiation/natural convection of participating medium in a cavity with a horizontal fin using LBM. Phys A Stat Mech Appl. 2018;509:1062–79.CrossRefGoogle Scholar
- 51.Taguchi G, Elsayed EA, Hsiang TC. Quality engineering in production systems, vol. 173. New York: McGraw-Hill; 1989.Google Scholar
- 52.Bhalla V, Khullar V, Tyagi H. Investigation of factors influencing the performance of nanofluid-based direct absorption solar collector using Taguchi method. J Therm Anal Calorim. 2019;135(2):1493–505.CrossRefGoogle Scholar
- 53.Taguchi G. Taguchi techniques for quality engineering. New York: Quality Resources; 1987.Google Scholar
- 54.Li X, Wang Z, Huang L. Study of vibration characteristics for orthotropic circular cylindrical shells using wave propagation approach and multivariate analysis. Meccanica. 2017;52(10):2349–61.CrossRefGoogle Scholar