Modeling of heat transfer at local convection over a vertical throughflow in a two-layered air-heat generating porous system
DOI:
https://doi.org/10.7242/1999-6691/2023.16.4.37Keywords:
local convection, air-porous system, heat generation, vertical throughflow, supercritical bifurcation, subcritical bifurcation, saddle-node bifurcation, heat transfer enhancementAbstract
Convective heat transfer in an air layer partially filled with a heat-generating granular porous medium is studied. There is a slow seepage of air through the layer in the vertical direction with a constant velocity. Equal temperatures are maintained at the outer solid permeable boundaries, and the heat source strength is constant within the porous sublayer and is proportional to the solid volume fraction. The permanent heat generation within the porous sublayer, combined with the vertical throughflow, causes a nonlinear thermal profile which is conducive for convection to occur. The Boussinesq approximation and Darcy's law are used to describe this convection. Numerical solution of the nonlinear convective problem is obtained on the basis of Newton's method. In the limiting case, the numerical data for the onset of convection are compared with the results of the earlier paper of the authors, where a linear stability theory and a method for constructing of the fundamental system of partial solution vectors have been applied, and with the data by other authors. The stationary regimes of local convection, which occurs in an “air – heat-generating porous medium-air” system over the basic vertical throughflow, and its effect on the heat transfer from the porous air sublayer with the growth of supercriticality are studied. It is shown that, depending on the velocity of the basic throughflow (the Peclet number), convection excitation can be both soft (due to supercritical pitchfork bifurcation) and hard (when the loss of stability of the basic throughflow is accompanied by subcritical pitchfork bifurcation that gives rise to an unstable secondary convective regime). This secondary regime is replaced by a stable tertiary convective regime with increasing supercriticality. It has been found that the total heat transfer rate for the upward basic throughflow exceeds that for the downward basic throughflow significantly, and that local convection at any direction of this throughflow increases the heat transfer rate in the system. An increase in the Nusselt number with the growth of supercriticality is recorded. However, a noticeable contribution of local convection to the total heat transfer is observed only when all values of the Pecklet number are negative and its positive values are lower than 2.
Downloads
References
Hu J.-T., Mei S.-J., Liu D., Zhao F.-Y., Wang H.-Q. Buoyancy driven heat and species transports inside an energy storage enclosure partially saturated with thermal generating porous layers. Int. J. Therm. Sci., 2018, vol. 126, pp. 38-55. https://doi.org/10.1016/j.ijthermalsci.2017.12.010
Lisboa K.M., Su J., Cotta R.M. Single domain integral transform analysis of natural convection in cavities partially filled with heat generating porous medium. Numer. Heat Tran., 2018, vol. 74, pp. 1068-1086. https://doi.org/10.1080/10407782.2018.1511141
Altukhov I.V., Ochirov V.D. Thermalphysic characteristics as the basis of calculation of the heating time constant of the sacchariferous root crops in the thermal processing processes. Vestnik KrasGAU – The Bulletin of KrasGAU, 2010, no. 4, pp. 134-139.
Bodrov V.I, Bodrov M.V. Teplomassoobmen v biologicheski aktivnykh sistemakh (teoriya sushki i khraneniya) [Heat and mass transfer in biologically active systems (theory of drying and storage)]. Nizhniy Novgorod, NNSUACE, 2013. 145 p.
Carr M. Penetrative convection in a superposed porous-medium-fluid layer via internal heating. J. Fluid Mech., 2004, vol. 509, pp. 305-329. https://doi.org/10.1017/S0022112004009413
Nield D.A., Bejan A. Convection in porous media. Springer, 2017. 988 p. https://doi.org/10.1007/978-3-319-49562-0
Yoon D.-Y., Kim D.-S., Choi C.K. Convective instability in packed beds with internal heat sources and throughflow. Korean J. Chem. Eng., 1998, vol. 15, pp. 341-344. https://doi.org/10.1007/BF02707091
Kuznetsov A.V., Nield D.A. The effect of vertical throughflow on the onset of convection induced by internal heating in a layered porous medium. Transp. Porous Med., 2013, vol. 100, pp. 101-114. https://doi.org/10.1007/s11242-013-0207-1
Kuznetsov A.V., Nield D.A. Local thermal non-equilibrium effects on the onset of convection in an internally heated layered porous medium with vertical throughflow. Int. J. Therm. Sci., 2015, vol. 92, pp. 97-105. https://doi.org/10.1016/j.ijthermalsci.2015.01.019
Suma S.P., Gangadharaiah Y.H., Indira R., Shivakumara I.S. Throughflow effects on penetrative convection in superposed fluid and porous layers. Transp. Porous Med., 2012, vol. 95, pp. 91-110. https://doi.org/10.1007/s11242-012-0034-9
Gangadharaiah Y.H. Influence of throughflow effects combined with internal heating on the onset of convection in a fluid layer overlying an anisotropic porous layer. JAMA, 2017, vol. 6, pp. 79-86. https://doi.org/10.1166/jama.2017.1129
Gangadharaiah Y.H., Nagarathnamma H., Hanumagowda B.N. Combined impact of vertical throughflow and gravity variance on Darcy-Brinkman convection in a porous matrix. International Journal of Thermofluid Science and Technology, 2021, vol. 8, 080303.
Yadav D. The onset of Darcy‐Brinkman convection in a porous medium layer with vertical throughflow and variable gravity field effects. Heat Transfer, 2020, vol. 49, pp. 3161-3173. https://doi.org/10.1002/htj.21767
Shvartsblat D.L. Steady convective motions in a plane horizontal fluid layer with permeable boundaries. Fluid Dyn., 1969, vol. 4(5), pp. 54-59. https://doi.org/10.1007/BF01015957
Gershuni G.Z., Zhukovitskii E.M. Convective stability of incompressible fluids. Jerusalem, Keter Publications, 1976, 330 p.
Chen F. Throughflow effects on convective instability in superposed fluid and porous layers. J. Fluid Mech., 1990, vol. 231, pp. 113-133. https://doi.org/10.1017/S0022112091003336
Chen F., Chen C.F. Onset of finger convection in a horizontal porous layer underlying a fluid layer. J. Heat Transfer, 1988, vol. 110, pp. 403-409. https://doi.org/10.1115/1.3250499
Lyubimov D.V., Muratov I.D. O konvektivnoi neustoichivosti v sloistoi sisteme [On convective instability in layered system]. Gidrodinamika – Hydrodynamics, 1977, no. 10, pp. 38-46.
Lyubimova T.P., Muratov I.D. Interaction of the longwave and finite-wavelength instability modes of convection in a horizontal fluid layer confined between two fluid-saturated porous layers. Fluids, 2017, vol. 2, 39. https://doi.org/10.3390/fluids2030039
Tsiberkin K. Porosity effect on the linear stability of flow overlying a porous medium. Eur. Phys. J. E, 2020, vol. 43, 34. https://doi.org/10.1140/epje/i2020-11959-6
Zubova N.A., Lyubimova T.P. Nonlinear convection regimes of a ternary mixture in a two-layer porous medium. Vychisl. mekh. splosh. sred – Computational Continuum Mechanics, 2021, vol. 14, no. 1, pp. 110-121. https://doi.org/10.7242/1999-6691/2021.14.1.10
Kolchanova E.A., Kolchanov N.V. Onset of solutal convection in layered sorbing porous media with clogging. Int. J. Heat Mass Tran., 2022, vol. 183, 122110. https://doi.org/10.1016/j.ijheatmasstransfer.2021.122110
Kolchanova E., Kolchanov N. Onset of internal convection in superposed air-porous layer with heat source depending on solid volume fraction: influence of different modeling. Acta Mech., 2022, vol. 233, pp. 1769-1788. https://doi.org/10.1007/s00707-022-03204-8
Kolchanova E., Sagitov R. Throughflow effect on local and large-scale penetrative convection in superposed air-porous layer with internal heat source depending on solid fraction. Microgravity Sci. Technol., 2022, vol. 34, 52. https://doi.org/10.1007/s12217-022-09971-2
Nikulin I.L., Perminov A.V. Matematicheskaya model’ konvektsii nikelevogo rasplava pri induktsionnom pereplave. Resheniye magnitnoy podzadach [The mathematical model of nickel melt convection in the induction melting. The solving of the magnetic subproblem]. Vestnik PNIPU. Mekhanika – PNRPU Mechanics Bulletin, 2013, no. 3, pp. 193-209.
Nehamkina O.A., Niculin D.A., Strelets M.Kh. Hierarchy of models of natural thermal convection of an ideal gas. High Temp., 1989, vol. 27, no. 6, pp. 883-892.
Ramazanov M.M. Conditions for the absence and occurrence of filtration convection in a compressible gas. J. Eng. Phys. Thermophy., 2014, vol. 87, pp. 541-547. https://doi.org/10.1007/s10891-014-1043-z
Kulacki F., Ramchandani R. Hydrodynamic instability in a porous layer saturated with a heat generating fluid. Wärme- und Stoffübertragung – Thermo and Fluid Dynamics, 1975, vol. 8, pp. 179-185. https://doi.org/10.1007/BF01681559
Horton C.W., Rogers F.T. Convection currents in a porous medium. J. Appl. Phys., 1945, vol. 16, pp. 367-370. https://doi.org/10.1063/1.1707601
Lapwood E.R. Convection of a fluid in a porous medium. Math. Proc. Camb. Phil. Soc., 1948, vol. 44, pp. 508-521. https://doi.org/10.1017/S030500410002452X
Katto Y., Matsuoka T. Criterion for onset of convective flow in a fluid in a porous medium. Int. J. Heat Mass Tran., 1967, vol. 10, pp. 297-309. https://doi.org/10.1016/0017-9310(67)90147-0
Glukhov A.F., Lyubimov D.V., Putin G.F. Konvektivnye dvizheniia v poristoi srede vblizi poroga neustoichivosti [Convective motions in a porous medium near the equilibrium instability threshold]. DAN SSSR, 1978, vol. 238, no. 3, pp. 549-551.
Glukhov A.F., Putin G.F. Eksperimental’noe issledovanie konvektivnykh struktur v nasyshchennoi zhidkost’iu poristoi srede vblizi poroga neustoichivosti mekhanicheskogo ravnovesiia [Experimental investigation of convection structures in a fluid-saturated porous medium near the mechanical equilibrium instability threshold]. Gidrodinamika – Hydrodynamics, 1999, no. 12, pp. 104-119.
Nouri-Borujerdi A., Noghrehabadi A.R., Rees D.A.S. Influence of Darcy number on the onset of convection in a porous layer with a uniform heat source. Int. J. Therm. Sci., 2008, vol. 47, pp. 1020-1025. https://doi.org/10.1016/j.ijthermalsci.2007.07.014
Carman P.C. Fluid flow through granular beds. Chem. Eng. Res. Des., 1997, vol. 75, pp. S32-S48. https://doi.org/10.1016/S0263-8762(97)80003-2
Torres Alvarez J.F. A study of heat and mass transfer in enclosures by phase-shifting interferometry and bifurcation analysis. Ecully, Ecole centrale de Lyon, 2014. 414 p.
Sagitov R.V., Sharifulin A.N. Bifurcations and stability of steady regimes of convective flows in an inclined rectangular cavity. Vychisl. mekh. splosh. sred – Computational Continuum Mechanics, 2018, vol. 11, no. 2, pp. 185-201. https://doi.org/10.7242/1999-6691/2018.11.2.15
Lobov N.I., Lyubimov D.V., Lyubimova T.P. Chislennye metody resheniia zadach teorii gidrodinamicheskoi ustoichivosti [Numerical methods for solving problems in the theory of hydrodynamic stability]. Perm, Izd-vo PGU, 2004. 101 p.
Downloads
Published
Issue
Section
License
Copyright (c) 1970 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.