Skip to main content
SpringerLink
Log in

Finite Element Simulation of Nonlinear Convective Heat and Mass Transfer in a Micropolar Fluid-Filled Enclosure with Rayleigh Number Effects

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

A mathematical model is presented to study the double-diffusive convective heat and mass transfer of a micropolar biofluid in a rectangular enclosure, as a model of transport phenomena in a bioreactor. The vertical walls of the enclosure are maintained at constant but different temperatures and concentrations. The conservation equations for linear momentum, angular momentum, energy and species concentration are formulated subject to appropriate boundary conditions and solved using both finite element and finite difference numerical techniques. Results are shown to be in excellent agreement between these methods. Several special cases of the flow regime are discussed. The distributions for streamline, isotemperature, isoconcentration and (isomicrorotation) are presented graphically for different Lewis number, buoyancy parameter, micropolar vortex viscosity parameter, gyration viscosity parameter, Rayleigh number, Prandtl number and micro-inertia parameter. Micropolar material parameters are shown to considerably influence the flow regime. The flow model has important applications in hybrid aerobic bioreactor systems exploiting rheological suspensions e.g. fermentation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Abbreviations

A :

Enclosure aspect ratio

B :

Micro-inertia density parameter

D :

Species (molecular) diffusivity

g :

Gravitational acceleration

\(G^{\prime }\) :

Dimensional micro-rotation component (angular velocity)

G :

Non-dimensional micro-rotation component (angular velocity)

\({{\varvec{he}}}\) :

x-Direction step-length in finite difference algorithm

\({{\varvec{H}}}^{\prime }\) :

Height of enclosure

\(K_{v}\) :

Eringen micropolar parameter

\([K^{mn}]\) :

Stiffness matrix in finite element domain

\({{\varvec{ke}}}\) :

y-Direction step-length in finite difference algorithm

Le :

Lewis number

\({{\varvec{L}}}^{\prime }\) :

Width of enclosure

M :

Angular velocity (microrotation) vector

N :

Buoyancy ratio

Nu :

Nusselt number

P :

Hydrodynamic pressure

Pr :

Prandtl number

R :

Eringen non-dimensional micropolar vortex viscosity parameter

Ra :

Rayleigh number

Sh :

Sherwood number

\(\varDelta S^{\prime }\) :

Concentration difference

\({{\varvec{S}}}_{{\varvec{h}}^{\prime }}\) :

Concentration at lower boundary of enclosure

\({{\varvec{S}}}_{{\varvec{1}}^{\prime }}\) :

Concentration at upper boundary of enclosure

\({{\varvec{S}}}_{{\varvec{0}}^{\prime }}\) :

Reference concentration at \({{\varvec{x}}^{\prime }}{={\varvec{0}}}, {{\varvec{y}}^{\prime }}{={\varvec{0}}}.\)

\(S^{\prime }\) :

Dimensional concentration

S :

Non-dimensional concentration

\({{\varvec{T}}}_{{\varvec{h}}^{\prime }}\) :

Temperature at vertical left wall of enclosure

\({{\varvec{Tc}}^{\prime }}\) :

Temperature at vertical right wall of enclosure

\({{\varvec{T}}}_{{\varvec{0}}^{\prime }}\) :

Reference temperature at \({{\varvec{x}}^{\prime }}{={\varvec{0}}}, {{\varvec{y}}^{\prime }}{={\varvec{0}}}.\)

\(\varDelta T^{\prime }\) :

Temperature difference

\(T^{\prime }\) :

Dimensional temperature

T :

Non-dimensional temperature

\(u^{\prime }\) :

Dimensional x\(^{\prime }\)-direction velocity

u :

Non-dimensional x\(^{\prime }\)-direction velocity

\(v^{\prime }\) :

Dimensional y\(^{\prime }\)-direction velocity

v :

Non-dimensional y\(^{\prime }\)-direction velocity

V :

Translational velocity vector

\(w_{i}\) :

Arbitrary test function in finite element model

\(x^{\prime }\) :

Coordinate parallel to base of enclosure (horizontal)

\(y^{\prime }\) :

Coordinate transverse to base of enclosure (vertical)

\(\alpha ^{*}\) :

Viscosity coefficient of micropolar fluid

\(\alpha \) :

Thermal diffusivity of micropolar fluid

\(\beta \) :

Viscosity coefficient of micropolar fluid

\(\gamma \) :

Viscosity coefficient of micropolar fluid

\(\lambda \) :

Viscosity coefficient of micropolar fluid (micropolar material parameter)

\(\mu \) :

Viscosity coefficient of micropolar fluid (dynamic viscosity)

\(\chi \) :

Viscosity coefficient of micropolar fluid

\(\rho \) :

Mass density of micropolar fluid

\(\beta _{S}\) :

Coefficient of species expansion

\(\beta _{T}\) :

Coefficient of thermal expansion

\({\varvec{\psi }}_{{\varvec{i}}}\) :

Linear interpolation function in finite element model

\({\varvec{\varOmega }}^{{\varvec{e}}}\) :

Rectangular element in finite element discretized domain

\(\nu \) :

Kinematic viscosity

\(\varphi ^{\prime }\) :

Dimensional stream function

\(\varphi \) :

Non-dimensional stream function

References

  1. Vant Riet, K., Tramper, J.: Basic Bioreactor Design. Marcel Dekker, New York (1991)

    Google Scholar 

  2. Galaction, A.I., Cascaval, D., Oniscu, C., Turnea, M.: Prediction of oxygen mass transfer coefficients in stirred bioreactors for bacteria, yeasts and fungus broths. Biochem. Eng. J. 20, 85–94 (2004)

    Article  Google Scholar 

  3. Mitchell, D.A., von Meien, O.F., Krieger, N.: Recent developments in modeling of solid-state fermentation: heat and mass transfer in bioreactors. Biochem. Eng. J. 13, 137–147 (2003)

    Article  Google Scholar 

  4. Van Kaam, R., Anne-Archard, D., Alliet Gaubert, M.: Rheological characterization of mixed liquor in a submerged membrane bioreactor. J. Membr. Sci. 317, 26–33 (2008)

    Article  Google Scholar 

  5. Ostrach, S.: Natural convection with combined driving forces. Phys. Chem. Hydrodyn. J. 1, 233–247 (1980)

    Google Scholar 

  6. Makham, B.I., Rosenberger, F.: Diffusive convection vapour transport across horizontal and inclined rectangular enclosures. J. Cryst. Growth 67, 241–254 (1984)

    Article  Google Scholar 

  7. Bergman, T.L., Incropera, F.P., Viskanta, R.: Correlation of mixed layer growth in a double-diffusive, salt-stratified system heated from below. ASME J. Heat Transf. 108, 206–211 (1986)

    Article  Google Scholar 

  8. Raganathan, P., Viskanta, R.: Natural convection in a square cavity due to combined driving forces. Numer. Heat Transf. 14, 35–59 (1988)

    Google Scholar 

  9. Nishimura, T., Imoto, T., Miyashita, H.: Occurrence and development of double-diffusive convection during solidification of a binary system. Int. J. Heat Mass Transf. 37, 1455–1464 (1994)

    Article  Google Scholar 

  10. Bejan, A., Trevisan, O.V.: Mass and heat transfer by high Rayleigh number convection in a porous medium heated from below. Int. J. Heat Mass Transf. 30, 2341–2356 (1987)

    Article  Google Scholar 

  11. Lin, K.W.: Unsteady natural convection heat and mass transfer in a saturated porous enclosure. Heat Mass Transf. J. 28, 49–56 (1993)

    Google Scholar 

  12. Mohamad, A.A., Bennacer, R.: Double-diffusion natural convection in an enclosure filled with saturated porous medium subjected to cross gradients. Int. J. Heat Mass Transf. 45, 3725–3740 (2002)

    Article  MATH  Google Scholar 

  13. Mohamad, A.A., Bennacer, R.: Natural convection in a confined saturated porous medium with horizontal temperature and vertical solutal gradients. Int. J. Therm. Sci. 40, 82–93 (2001)

    Article  Google Scholar 

  14. Khanafer, K., Chamkha, A.J.: Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. Int. J. Heat Mass Transf. 42, 2465–2481 (1999)

    Article  MATH  Google Scholar 

  15. Khanafer, K., Vafai, K.: Double-diffusive mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. Numer. Heat Transf. 42, 465–486 (2002)

    Article  Google Scholar 

  16. Eringen, A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)

    MathSciNet  Google Scholar 

  17. Jena, S.K., Bhattacharyya, S.P.: The effect of microstructure on the thermal convection in a rectangular box of fluid heated from below. Int. J. Eng. Sci. 24, 69–76 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, C.K., Hsu, T.H.: Natural convection of micropolar fluids in an enclosure. In: ASME/JSME Thermal Engineering Conference Proceedings, vol. 1, pp. 199–205. Tokyo, Japan (1991)

  19. Hsu, T.H., Chen, C.K.: Natural convection of micropolar fluids in a rectangular enclosure. Int. J. Eng. Sci. 34(4), 407–415 (1996)

    Article  MATH  Google Scholar 

  20. Chiu, C.P., Shich, J.Y., Chen, W.R.: Transient natural convection of micropolar fluids in concentric spherical annuli. Acta Mech. 132, 75–92 (1999)

    Article  MATH  Google Scholar 

  21. Srinivas, J., Ramana Murthy, J.V.: Second law analysis of the flow of two immiscible micropolar fluids between two porous beds. J. Eng. Thermophys. 25(1), 126–142 (2016)

    Article  Google Scholar 

  22. Ramana Murthy, J.V., Srinivas, J., Sai, K.S.: Flow of immiscible micropolar fluids between two porous beds. J. Porous Media 17(4), 287–300 (2014)

    Article  Google Scholar 

  23. Eringen, A.C.: Micro-Continuum Field Theories: Volume II- Fluid Media, vol. II. Springer, New York (2001)

    Google Scholar 

  24. Reddy, J.N.: An Introduction to the Finite Element Method. McGraw-Hill Book Co., New York (1985)

    MATH  Google Scholar 

  25. Bég, O.A., Bég, T.A., Bhargava, R., Rawat, S., Tripathi, D.: Finite element study of pulsatile magneto-hemodynamic non-Newtonian flow and drug diffusion in a porous medium channel. J. Mech. Med. Biol. 12(4), 12500811–125008126 (2012)

    Article  Google Scholar 

  26. Rana, P., Bhargava, R., Bég, O.A.: Finite element modeling of conjugate mixed convection flow of \(\text{ Al }_{2}\text{ O }_{3}\) -water nanofluid from an inclined slender hollow cylinder. Phys. Scr. 88, 15 (2013)

    Google Scholar 

  27. Rajesh, V., Bég, O.A.: MHD transient nanofluid flow and heat transfer from a moving vertical cylinder with temperature oscillation. Comput. Therm. Sci. 6, 439–450 (2014)

    Article  Google Scholar 

  28. Bhargava, R., Sharma, S., Bég, O.A., Zueco, J.: Finite element study of nonlinear two-dimensional deoxygenated biomagnetic micropolar flow. Commun. Nonlinear Sci. Numer. Simul. 15, 1210–1233 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Andersen, D.A., et al.: Computational Fluid Mechanics and Heat Transfer. Hemisphere, Washington (1984)

    Google Scholar 

  30. Sohail, A., Wajid, H.A., Rashidi, M.M.: Numerical modeling of capillary-gravity waves using the phase field method. Surf. Rev. Lett. 21(03), 1450036 (2014)

    Article  Google Scholar 

  31. Sohail, A., Uddin, M.J., Rashidi, M.M.: Numerical study of free convective flow of a nanofluid over a chemically reactive porous flat vertical plate with a second-order slip model. ASCE J. Aerosp. Eng. 29(2), 04015047 (2015)

  32. Bég, O.A., Bhargava, R., Rawat, S., Takhar, H.S., Bég, T.A.: A study of buoyancy-driven dissipative micropolar free convection heat and mass transfer in a Darcian porous medium with chemical reaction. Nonlinear Anal. Model. Control J. 12(2), 157–180 (2007)

    MATH  Google Scholar 

  33. Rashidi, M.M., Ferdows, M., Uddin, Md Jashim, Bég, O.A., Rahimzadeh, N.: Group theory and differential transform analysis of mixed convective heat and mass transfer from a horizontal surface with chemical reaction effects. Chem. Eng. Commun. 199(8), 1012–1043 (2012)

    Article  Google Scholar 

  34. Bég, O.A., Rashidi, M.M., Keimanesh, M., Bég, T.A.: Semi-numerical modelling of “chemically-frozen” combusting buoyancy-driven boundary layer flow along an inclined surface. Int. J. Appl. Math. Mech. 9(1), 1–16 (2013)

    Google Scholar 

  35. Bég, O.A., Motsa, S.S., Islam, M.N., Lockwood, M.: Pseudo-spectral and variational iteration simulation of exothermically-reacting Rivlin–Ericksen viscoelastic flow and heat transfer in a rocket propulsion duct. Comput. Therm. Sci. 6(1), 1–12 (2014). (12 pages)

    Article  Google Scholar 

  36. Rajesh, V., Bég, O.A., Sridevi, Ch., Jonah Philliph, K.: Finite element analysis of unsteady MHD free convective laminar boundary-layer accelerated dissipative flow with uniform suction and chemical reaction. Int. J. Energy Technol. 6, 1–10 (2014)

    Article  Google Scholar 

  37. Uddin, M.J., Bég, O.A., Aziz, A., Ismail, A.I.M.: Group analysis of free convection flow of a magnetic nanofluid with chemical reaction. Math. Prob. Eng. 1–11 (2015). doi:10.1155/2015/621503. Article ID 621503

  38. Kiran, G.R., Radhakrishnamacharya, G., Bég, O.A.: Peristaltic flow and hydrodynamic dispersion of a reactive micropolar fluid: simulation of chemical effects in the digestive process. J. Mech. Med. Biol. (2016). doi:10.1142/S0219519417500130

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Roorkee, India

    R. Bhargava & S. Sharma

  2. Fluid Mechanics, Department of Civil Engineering, Indian Institute of Technology, Roorkee, India

    P. Bhargava

  3. Fluid Mechanics, Spray Research Group, School of Computing, Science and Engineering, University of Salford, G77, Newton Building, Manchester, M54WT, UK

    O. Anwar Bég & A. Kadir

Authors
  1. R. Bhargava
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Bhargava
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. O. Anwar Bég
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Kadir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. Anwar Bég.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhargava, R., Sharma, S., Bhargava, P. et al. Finite Element Simulation of Nonlinear Convective Heat and Mass Transfer in a Micropolar Fluid-Filled Enclosure with Rayleigh Number Effects. Int. J. Appl. Comput. Math 3, 1347–1379 (2017). https://doi.org/10.1007/s40819-016-0180-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-016-0180-9

Keywords

Search

Navigation