# MHD free convection heat transfer of a water–Fe_{3}O_{4} nanofluid in a baffled C-shaped enclosure

## Abstract

In this paper, the effect of a baffle on free convection heat transfer of a water–Fe_{3}O_{4} nanofluid in a C-shaped enclosure in the presence of a magnetic field is investigated numerically. The enclosure is subjected to a constant magnetic field. The vertical wall on the left side is maintained at a constant hot temperature of *T*_{h}, and the right one is kept at a constant cold temperature of *T*_{c}. The rest of the walls are insulated. The governing equations are discretized by the control volume method and solved simultaneously by the SIMPLE algorithm. The numerical results show very good agreement with other published works. The results indicate that by increasing the enclosure’s aspect ratio, the Nusselt number is increased. It is also found that the volume fraction of nanoparticles can be raised in order to achieve increased cooling in the enclosure. By increasing the aspect ratio, the effect of the nanoparticles on the enhancement of the Nusselt number is more pronounced. Also, the maximum effect of the baffle on the heat transfer is seen at the bottom of the hot wall. Generally, increasing the baffle length produces increases in the Nusselt number. The maximum cooling level is occurred for AR = 0.7 and *B*_{f} = 0.2.

## Keywords

Magnetic field Nanofluid Free convection C-shaped enclosure Baffle## List of symbols

*a*Baffle length

- AR
Aspect ratio,

*H*/*L**B*_{0}Magnetic field strength,

*T**B*_{f}Dimensionless baffle length,

*a*/*L**C*_{p}Specific heat at constant pressure (J kg-K

^{−1})*g*Gravitational acceleration (m s

^{−2})*H*Length of heat source (m)

*Ha*Hartmann number, \(B_{0} L\sqrt {\sigma_{\text{f}} /\rho_{\text{f}} \nu_{\text{f}} }\)

*k*Thermal conductivity (Wm

^{−1}K^{−1})*L*Length of cavity (m)

*Nu*Local Nusselt number

*Nu*_{m}Average Nusselt number of heat source

*p*Fluid pressure (Pa)

*P*Dimensionless pressure, \(pH/\rho_{\text{nf}} \alpha_{\text{f}}^{2}\)

*Pr*Prandtl number, \(\nu_{\text{f}} /\alpha_{\text{f}}\)

*Ra*Rayleigh number, \(g\beta_{\text{f}} \left( {T_{\text{h}} - T_{\text{c}} } \right)H^{3} /\alpha_{\text{f}} \vartheta_{\text{f}}\)

*T*Temperature (K)

*T*_{c}Cold wall temperature (K)

*T*_{h}Heated wall temperature (K)

*u*,*v*Velocity components in the

*x*,*y*directions (m s^{−1})*U*,*V*Dimensionless velocity components, \(u/v_{0} ,v/v_{0}\)

*x*,*y*Cartesian coordinates (m)

*X*,*Y*Dimensionless coordinates,

*x*/*L*,*y*/*L*

## Greek symbols

*α*Thermal diffusivity,

*k*/*ρc*_{p}(m^{2}s^{−1})*β*Thermal expansion coefficient (K

^{−1})*φ*Solid volume fraction

*σ*Effective electrical conductivity (μS cm

^{−1})*κ*_{b}Boltzmann constant (J K

^{−1})*θ*Dimensionless temperature, \(\left( {T - T_{\text{c}} } \right)/\left( {T_{\text{h}} - T_{\text{c}} } \right)\)

*μ*Dynamic viscosity (N s m

^{2})*ν*Kinematic viscosity (m

^{2}s^{−1})*ρ*Density (kg m

^{−3})

## Subscripts

- c
Cold

- eff
Effective

- f
Pure fluid

- h
Hot wall

- m
Average

- nf
Nanofluid

- s
Nanoparticle

## References

- 1.Estelle P, Mahian O, Mare T, Öztop HF. Natural convection of CNT water-based nanofluids in a differentially heated square cavity. J Therm Anal Calorim. 2017;128:1765–70.CrossRefGoogle Scholar
- 2.Moshizi SA, Malvandi A. Different modes of nanoparticle migration at mixed convection of Al
_{2}O_{3}–water nanofluid inside a vertical microannulus in the presence of heat generation/absorption. J Therm Anal Calorim. 2016;126:1947–62.CrossRefGoogle Scholar - 3.Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S. A review of the applications of nanofluids in solar energy. Int J Heat Mass Transf. 2013;57:582–94.CrossRefGoogle Scholar
- 4.Amraqui S, Mezrhab A, Abid C. Computation of coupled surface radiation and natural convection in an inclined «T» form cavity. Energy Convers Manag. 2011;52(2):1166–74.CrossRefGoogle Scholar
- 5.Chamkha AJ, Ismael MA. Conjugate heat transfer in a porous cavity heated by a triangular thick wall. Numer Heat Transf Part A Appl. 2013;63:144–58.CrossRefGoogle Scholar
- 6.El Alami M, Najam M, Semma E, Oubarra A, Penot F. Chimney effect in a “T” form cavity with heated isothermal blocks: the blocks height effect. Energy Convers Manag. 2004;45(20):3181–91.CrossRefGoogle Scholar
- 7.Bilgen E. Natural convection in cavities with a thin fin on the hot wall. Int J Heat Mass Transf. 2005;48(17):3493–505.CrossRefGoogle Scholar
- 8.Shenoy A, Sheremet M, Pop I. Convective flow and heat transfer from wavy surfaces: viscous fluids, porous media and nanofluids. New York: CRC Press, Taylor & Francis Group; 2016.CrossRefGoogle Scholar
- 9.Kakaç S, Pramuanjaroenkij A. Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transf. 2009;52:3187–96.CrossRefGoogle Scholar
- 10.Groşan T, Sheremet MA, Pop I. Heat transfer enhancement in cavities filled with nanofluids. In: Minea AA, editor. Advances in heat transfer fluids: from numerical to experimental techniques. New York: CRC Press, Taylor & Francis; 2017. p. 267–84.Google Scholar
- 11.Eastman JA, Choi SU, Li S, Yu W, Thompson LJ. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett. 2001;78(6):718–20.CrossRefGoogle Scholar
- 12.Das SK, Choi SUS, Yu W, Pradeep Y. Nanofluids: science and technology. New Jersey: Wiley; 2008.Google Scholar
- 13.Mahmoudi AH, Shahi M, Raouf AHA. Ghasemian numerical study of natural convection cooling of horizontal heat source mounted in a square cavity filled with nanofluid. Int. Commun Heat Mass Transf. 2010;37(8):1135–41.CrossRefGoogle Scholar
- 14.Jou RY, Tzeng SC. Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures. Int Commun Heat Mass Transf. 2006;33(6):727–36.CrossRefGoogle Scholar
- 15.Khanafer K, Vafai K, Lightstone M. Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transf. 2003;46(19):3639–53.CrossRefGoogle Scholar
- 16.Ahmed SE, Rashad AM. Natural convection of micropolar nanofluids in a rectangular enclosure saturated with anisotropic porous media. J Porous Media. 2016;19(8):737–50.CrossRefGoogle Scholar
- 17.Rashad AM, Gorla R, Mansour MA, Ahmed SE. Magnetohydrodynamic effect on natural convection in a cavity filled with porous medium saturated with nanofluid. J Porous Media. 2017;20(4):363–79.CrossRefGoogle Scholar
- 18.Sivasankaran S, Mansour MA, Rashad AM, Bhuvaneswari M. MHD mixed convection of Cu–water nanofluid in a two-sided lid-driven porous cavity with a partial slip. Numer Heat Transf Part A. 2016;70(12):1356–70.CrossRefGoogle Scholar
- 19.Mansour MA, Ahmed SE, Rashad AM. MHD natural convection in a square enclosure using nanofluid with the influence of thermal boundary conditions. J Appl Fluid Mech. 2016;9(5):2515–25.Google Scholar
- 20.Rashad AM, Sivasankaran S, Mansour MA, Bhuvaneswari M. Magneto-convection of nanofluids in a lid-driven trapezoidal cavity with internal heat generation and discrete heating. Numer Heat Transf Part A Appl. 2017;71(12):1223–34.CrossRefGoogle Scholar
- 21.Rahimi A, Kasaeipoor A, Malekshah EH, Amiri A. Natural convection analysis employing entropy generation and heatline visualization in a hollow L-shaped cavity filled with nanofluid using lattice Boltzmann method—experimental thermo-physical properties. Physica E. 2018;97:92–7.Google Scholar
- 22.Gorla RSR, Chamkha AJ. Natural convective boundary layer flow over a non-isothermal vertical plate embedded in a porous medium saturated with a nanofluid. Nanoscale Microscale Therm. 2011;15:81–94.CrossRefGoogle Scholar
- 23.Ismael MA, Armaghani T, Chamkha AJ. Conjugate heat transfer and entropy generation in a cavity filled with a nanofluid-saturated porous media and heated by a triangular solid. J Taiwan Inst Chem Eng. 2016;59:138–51.CrossRefGoogle Scholar
- 24.Armaghani T, Ismael MA, Chamkha AJ. Analysis of entropy generation and natural convection in an inclined partially porous layered cavity filled with a nanofluid. Can J Phys. 2017;95:238–52.CrossRefGoogle Scholar
- 25.Armaghani T, Kasaeipoor A, Alavi N, Rashidi MM. Numerical investigation of water–alumina nanofluid natural convection heat transfer and entropy generation in a baffled L-shaped cavity. J Mol Liq. 2016;223:243–51.CrossRefGoogle Scholar
- 26.Snoussi L, Ouerfelli N, Chesneau X, Chamkha AJ, Belgacem FBM, Guizani A. Natural convection heat transfer in a nanofluid filled U-shaped enclosures: numerical investigations. Heat Transf Eng. 2017.Google Scholar
- 27.Rashad AM, Armaghani T, Chamkha AJ, Mansour MA. Entropy generation and MHD natural convection of a nanofluid in an inclined square porous cavity: effects of a heat sink and source size and location. Chin J Phys. 2018;56(1):193–211.CrossRefGoogle Scholar
- 28.Chamkha AJ, Rashad AM, Mansour MA, Armaghani T, Ghalambaz M. Effects of heat sink and source and entropy generation on MHD mixed convection of a Cu–water nanofluid in a lid-driven square porous enclosure with partial slip. Phys Fluids. 2017;29(5):2001–22.CrossRefGoogle Scholar
- 29.Chamkha AJ, Rashad AM, Armaghani T, Mansour MA. Effects of partial slip on entropy generation and MHD combined convection in a lid-driven porous enclosure saturated with a Cu–water nanofluid. J Therm Anal Calorim. 2017.Google Scholar
- 30.Kasaeipoor A, Ghasemi B, Aminossadati SM. Convection of Cu–water nanofluid in a vented T-shaped cavity in the presence of magnetic field. Int J Therm Sci. 2015;94:50–60.CrossRefGoogle Scholar
- 31.Makulati N, Kasaeipoor A, Rashidi MM. Numerical study of natural convection of a water–alumina nanofluid in inclined C-shaped enclosures under the effect of magnetic field. Adv Powder Technol. 2016;27(2):661–72.CrossRefGoogle Scholar
- 32.Chamkha AJ, Ismael MA, Kasaeipoor A, Armaghani T. Entropy generation and natural convection of CuO–water nanofluid in C-shaped cavity under magnetic field. Entropy. 2016;18(50):1–18.Google Scholar
- 33.Bergman TL. Effect of reduced specific heats of nanofluids on single phase, laminar internal forced convection. Int J Heat Mass Transf. 2009;52:1240–4.CrossRefGoogle Scholar
- 34.Maxwell JC. A treatise on electricity and magnetism. Oxford: Clarendon Press; 1881.Google Scholar
- 35.Brinkman HC. The viscosity of concentrated suspensions and solutions. J Chem Phys. 1952;20(4):571.CrossRefGoogle Scholar
- 36.Patel HE, Anoop KB, Sundararajan T, Das SK. A micro-convection model for thermal conductivity of nanofluids. Int Heat Transf Conf 13. 2006. https://doi.org/10.1615/IHTC13.p8.240.
- 37.Santra AK, Sen S, Chakraborty N. Study of heat transfer due to laminar flow of copper–water nanofluid through two isothermally heated parallel plates. Int J Therm Sci. 2009;48(2):391–400.CrossRefGoogle Scholar
- 38.Sheikholeslami M, Rashidi MM. Effect of space dependent magnetic field on free convection of Fe
_{3}O_{4}–water nanofluid. J Taiwan Inst Chem Eng. 2015;56:6–15.CrossRefGoogle Scholar - 39.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
- 40.Mahmoodi M, Hashemi SM. Numerical study of natural convection of a nanofluid in C-shaped enclosures. Int J Therm Sci. 2012;55:76–89.CrossRefGoogle Scholar