STABILITY OF THERMOSOLUTAL MIXED CONVECTION IN A VERTICAL PIPE FILLED WITH POROUS MEDIUM

Abstract

The dynamics of heat and mass transfer for flows induced by temperature and concentration fields and external pressure gradient (i.e. double diffusive mixed convective flows) are expected to be very different from those driven by the external pressure gradient and temperature field solely. The theory of double diffusive mixed convection in porous media has many unsolved problems in engineering and applied sciences ranging from contaminant transport [50], geophysics: such as hot springs [148], packed bed reactors [4], hydrothermal vents [37], petroleum engineering [68] to marine science [54]. Among others, the problem related to hot springs or hydrothermal vents, where molten rocks from the lower layers (magma) transport hot fluid combined with many chemical compounds gives rise to hydrothermal activities mediated by a complex convective phenomenon, is of our special interest. In the Indian subcontinent several hot springs occur. Examples are those in the water reservoirs of Surajkund in Hazaribag district of Jharkhand with a maximum temperature of approximately 87 degree centigrade, the Taptakund in Badrinath (Uttarakhand), seven hot springs in Bakreswar(West Bengal). Recent long term studies show that the ecosystem of the surrounding zones (permeable porous medium) as well as the water level have changed drastically. The reservoir bottom in these cases consist of permeable porous sand. Measurements show that the temperature in the vents and hot springs appearing therein decreases upward. Consequently, the solute concentration is assumed to decrease from bottom to top. i ii Often hydrothermal vents or hot springs appear in a certain horizontal distances to each other. The model we propose in this present thesis is designed to mimic the exchange of fluid, heat and concentration in the sediment layers lying in between. Keeping in mind the importance of above studies, we thought of analyzing a geometry which would be a simplified representation of many hot springs, which in turn gives rise to a vertical pipe flow filled with porous medium. Recently, Kumar and Bera [70] have studied the non-Darcy fully developed mixed convection in a vertical pipe filled with porous media in which they found that velocity profile posses point of inflection (flow separation) beyond a threshold value of Rayleigh number in buoyancy assisted (buoyancy opposed) case. Their study was restricted to the situation when both the solid porous matrix and saturated fluid were in local thermal equilibrium state. Actually, the solid porous matrix may have a different temperature from that of the saturating fluid [61], this being understood in terms of averages over representative elementary volumes. Thus hot fluid can flow into a cold, relatively insulating porous matrix, and there will exist a difference in the average local temperature of the two phases which will take time to reduce to values where one could say that the phases are in local thermal equilibrium state (LTE). Therefore in this situation, assumption of local thermal non-equilibrium (LTNE) state will be much more appropriate than LTE. Definitely, physics of the problem will be affected by consideration of LTNE. Beside this, it is also important to know how the physics will be modulated when the two buoyancy forces: due to temperature and solute are also taken into system. In order to investigate the impact of LTNE as well as to answer the above query, an attempt has been taken in the present thesis by incorporating solute transport along with validity of LTNE assumption in previous work of Kumar and Bera [70]. The present thesis is compiled in 6 chapters. Chapter 1 is introductory and contains a brief outline of the fundamentals of iii fluid flow and heat as well as mass transfer in porous media that includes possible models for both equilibrium and non-equilibrium hypothesis and a brief description of hydrodynamics stability theory. The brief outline about the numerical method which is adopted for the numerical solution is also given in this chapter. Chapter 2 highlights the review of literature, mainly related to (i) mixed convection in vertical system, (ii) double-diffusive convection in vertical system and (iii) convection in porous media under LTNE state. In Chapter 3, the fully developed double-diffusive mixed convection in a vertical pipe under local thermal non-equilibrium state has been investigated. The non-Darcy Brinkman-Forchheimer extended model has been used and solved numerically by spectral collocation method. Special attention is given to understand the effect of buoyancy ratio (N) and thermal non-equilibrium parameters: inter phase heat transfer coefficient (H) as well as porosity scaled thermal conductivity ratio (g) on the flow profiles as well as on the rates of heat and solute transfer. Judged from the influence of buoyancy ratio on velocity profiles, when both the buoyancy forces: thermal as well as solutal are in favor of each other, it has been found that for given any value of H and N equal to 10 as well as 100 the basic velocity profile shows back flow in small sub domain of the domain of the flow. When two buoyancy forces are opposite to each other (RaT = -1000), velocity profile possesses a kind of distortion in which the number of zeroes increases on increasing N. Corresponding variation of heat transfer rate in the (N, Nuf )-plane shows a sinusoidal pattern. The flow separation in the flow profile dies out on increasing H for N = 0. It has also been found that for each N, when N < 0.7, there exists a minimum value of H such that the velocity profile becomes free from flow separation. Influence of H on the profiles of solid temperature as well as solute in both situations is similar. Overall, the impact of LTNE parameters, specially g, on the heat transfer rate of double-diffusive convection is not straight forward. iv Chapter 4, contains the linear stability of the double diffusive mixed convective flow in porous media using a non-Darcy Brinkman-Forchheimer extended model. The stability of the basic flow is controlled by the governing parameters: Reynolds number, Rayleigh thermal number, Rayleigh solutal number, Prandtl number, Schimdt number, Darcy number, porosity and specific heat capacity ratio. The instability mechanism of basic flow has been investigated numerically using the spectral collocation technique. The linear stability is performed using a wide range of Darcy numbers, ranging from 10¡8 to 10¡5. The instability boundary curve shows three distinct regimes (i) Rayleigh- Taylor (R-T), (ii) log-log non linear variation and (iii) log-log linear variation. The similar kind of regimes are also demonstrated by Thangam et al. [136], Kerr and Tang [64], and Young and Rosner [160] while studying cross diffusion in a vertical slot of purely viscous fluid. Recently Bera and Kumar [16] have reported this phenomena in porous media, but their study is restricted in channel. Simulations for the secondary flow profile are also demonstrated at the critical state of all the three regimes. The above study indicates that the first azimuthal mode is least stable mode except in few cases. A direct link is found between the critical RaT in the Rayleigh-Taylor mode and Darcy number which is given by -RaTDa = 2.467. A hyperbolic relationship between RaT and Da is found at Re = 1000 and is given by RaTDa =7.7 x 10¡4 for first azimuthal mode and it is shifted to 5.0 x 10¡4 for the zero azimuthal mode. Some other interesting results are also obtained during the above study while simulating the secondary flow profile of isothermal and isosolutal lines. In Chapter 5, the linear stability of the double diffusive mixed convective flow in porous media under local thermal non-equilibrium model is investigated numerically. This chapter is a continuation part of the chapter 2. It is also an interesting extension of the chapter 3 in which the LTNE model is introduced while performing the linear stability analysis. The objective of this chapter is to understand that how the physics of the flow dynamics is affected after introducing the v coupled energy equations, one for fluid phase and the other for solid phase. Here, the stability of this flow is controlled by the governing parameters: Reynolds number, Rayleigh thermal number, Buoyancy ratio, Prandtl number, Schimdt number, Darcy number, interphase heat transfer coefficient, porosity scaled conductivity ratio, porosity and specific heat capacity ratio. It has been found that the first azimuthal mode is always least stable in most of the cases except in some cases depending on the value of media permeability as well as Reynolds number. The stability analysis indicates that for a given Reynolds number (Re), the fully developed base flow is highly unstable for high buoyancy force (N), whereas, the interphase heat transfer coefficient (H) stabilizes the base flow. Both pattern as well as magnitude of the secondary isotherm and isosolutal profile varies significantly on changing media permeability, interface heat transfer coefficient (H) and buoyancy force (N). Finally, Chapter 6 presents the summary and conclusions of this thesis and the possible directions of future work.