Solutal Rayleigh Number Research Articles

Control of convective dissolution by chemical reactions: general classification and application to CO(2) dissolution in reactive aqueous solutions.

In partially miscible two-layer systems within a gravity field, buoyancy-driven convective motions can appear when one phase dissolves with a finite solubility into the other one. We investigate the influence of chemical reactions on such convective dissolution by a linear stability analysis of a reaction-diffusion-convection model. We show theoretically that a chemical reaction can either enhance or decrease the onset time of the convection, depending on the type of density profile building up in time in the reactive solution. We classify the stabilizing and destabilizing scenarios in a parameter space spanned by the solutal Rayleigh numbers. As an example, we experimentally demonstrate the possibility to enhance the convective dissolution of gaseous CO_{2} in aqueous solutions by a classical acid-base reaction.

Double diffusive convection in a couple stress fluid saturated porous layer with internal heat source

Abstract The onset of double diffusive convection in a couple stress fluid saturated porous layer with an internal heat source is studied using linear and weak nonlinear stability analyses. The linear analysis is based on the classical normal mode technique. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. In the present context, double diffusive convection is of particular interest in the study of extraction of metals from ores where a mushy layer is formed during solidification of a metallic alloy. Further, internal heating of the system is used as an external means to influence the transport process, thereby controlling the quality and structure of the resulting solid. The expressions for stationary, oscillatory and finite-amplitude Rayleigh number are obtained as a function of governing parameters such as internal Rayleigh number, couple stress parameter, solute Rayleigh number, Darcy-Prandtl number, normalized porosity and Lewis number and their effects on the stability of a system are shown graphically. It is observed that the onset of both stationary and oscillatory convection is advanced by the internal Rayleigh number. The nonlinear analysis is based on the truncated representation of Fourier series which provides the quantification of heat and mass transfer. Heat transport is reinforced while mass transport is suppressed by the internal Rayleigh number. The transient behavior of Nusselt and Sherwood numbers is studied by solving numerically a fifth order Lorenz type system using Runge–Kutta method. Some known results are recovered as the particular cases of the present study.

Double Diffusive Convection in a Layer of Maxwell Viscoelastic Fluid in Porous Medium in the Presence of Soret and Dufour Effects

Double diffusive convection in a horizontal layer of Maxwell viscoelastic fluid in a porous medium in the presence of temperature gradient (Soret effects) and concentration gradient (Dufour effects) is investigated. For the porous medium Darcy model is considered. A linear stability analysis based upon normal mode technique is used to study the onset of instabilities of the Maxwell viscolastic fluid layer confined between two free-free boundaries. Rayleigh number on the onset of stationary and oscillatory convection has been derived and graphs have been plotted to study the effects of the Dufour parameter, Soret parameter, Lewis number, and solutal Rayleigh number on stationary convection.

Linear Stability Analysis of Double Diffusive Convection in a Horizontal Sparsely Packed Rotating Anisotropic 1 Porous Layer in Presence of Soret Effect

In this paper, we have investigated theoretically the effect of Soret parameter on the onset of double diffusive rotating anisotropic convection in a horizontal sparsely packed porous layer using linear stability theory which is based on the usual normal mode technique. The Brinkman model that includes the Coriolis term is employed for the momentum equation. The effect of anisotropy parameters, Soret parameter, solute Rayleigh number, Taylor number, Lewis number, Darcy and Darcy Prandtl number on stationary and oscillatory convection is shown graphically.

Effects of geochemical reaction on double diffusive natural convection of CO2 in brine saturated geothermal reservoir

In order for mitigation of greenhouse gas emissions, as we know, the storage of CO2 in deep geological formations is considered as a promising option. Besides convective mixing an understanding of geochemical reaction that affects the long term storage process in deep saline aquifers is of great importance. In this study we numerically investigate double diffusive natural convection of CO2 in brine saturated geothermal reservoir which is anisotropic in permeability variations, impervious from the sides, and is open to CO2 at the top. Our primary objective is to understand effects of geochemical reactions of different rates and orders on density driven natural convection of CO2 due to concentration and geothermal gradients. We present propagation of CO2 plumes over long period of time by analyzing different combination of problem parameters: Damkohler number (0.01⩽Da⩽105) with variation of 1st, 2nd, and 3rd order reactions; solutal Rayleigh number (500⩽Ras⩽2000); the buoyancy ratio (2⩽N⩽100); Dykstra–Parsons coefficient (0.35⩽Vdp⩽0.85); and fixed Lewis and Prandtl numbers. Reaction order is set from relevant stoichiometric ratio. In each case results are quantified in term of percentage of CO2 deposition. It is found that mineral interactions make traceable difference of depositions as reaction rate increases, especially when Da>103. In one hand, compared to very minimum geochemical interaction (Da∼0.01) the strong reaction effects (Da∼104) can make difference of more than 5% in the period of 500years of trapping. On the other hand, at a fixed equivalent Da reaction order also makes substantial distinction as deposition time passes. Heterogeneity plays a vital role. Geothermal gradient has very negligible effect, however only after very long time (>500years).

Effect of Internal Heating on Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Medium

The effect of internal heat source on double diffusive convection in a couple stress fluid saturated horizontal anisotropic porous layer was studied analytically by performing linear and nonlinear stability analyses. For linear stability analysis, normal mode technique was used, whereas for nonlinear stability analysis a minimal representation of Fourier series upto two terms was used. Effects of anisotropic parameter, couple stress parameter, solute Rayleigh number, Vadasz number, Lewis number, internal heat source parameter on stationary, oscillatory and finite amplitude convection were obtained and shown graphically. Also the graphs for heat and mass transport, streamlines, isotherms, isohalines for steady and unsteady cases were drawn. It was found that internal heat source parameter had destabilizing effect on all modes of convection. Heat transport increased and mass transport decreased with the increase in internal heat source parameter.

Double diffusive natural convection of CO2 in a brine saturated geothermal reservoir: Study of non-modal growth of perturbations and heterogeneity effects

In a recent study (Islam et al., 2013) double diffusive natural convection of CO2 in brine saturated homogeneous permeable geothermal reservoir was investigated. Here we have shown the study of heterogeneous reservoirs. Firstly, we present the non-modal stability theory to analyze time-dependent double diffusive convection. Linear stability analysis is performed and the maximum amplification of perturbations optimized over the entire space of initial perturbations is examined. These provide us onset time of convection (or the critical time) and the structure of the most-amplified perturbations. Secondly, we investigate numerical simulation of an anisotropic and layered reservoir with permeability variations, which is impervious from the sides, and is open to CO2 at the top. We show the propagation of CO2 plumes over long period of time by analyzing different combination of problem parameters: solutal Rayleigh number (100≤Ras≤10,000), the buoyancy ratio (2≤N≤100), Dykstra-Parsons coefficient (0≤Vdp≤0.85) and a fixed Lewis number. How permeability variation in vertical (kz) to horizontal (kx) direction affects the CO2 dissolution is discussed for different ratios of kz/kx (0.1≤kz/kx≤0.8). Results of different cavity aspect ratios (0.5≤A≤2) are also presented. It is found that heterogeneous porous reservoir renders better mass transfer of CO2 than the homogeneous one. CO2 dissolution is enhanced with increasing both Ras and Vdp. N, which represents geothermal effect, has minor impact on overall dissolution process. The results have implications in enhanced oil recovery and CO2 based geothermal systems.

Effects of Geochemical Reaction on Double Diffusive Natural Convection of CO2 in Brine Saturated Geothermal Reservoir

Abstract In order for mitigation of greenhouse gas emissions, as we know, the storage of CO 2 in deep geological formations is considered as a promising option. Besides convective mixing an understanding of geochemical reaction that affects the long term storage process in deep saline aquifers is of great importance. In this study we numerically investigate double diffusive natural convection of CO 2 in brine saturated geothermal reservoir which is anisotropic in permeability variations, impervious from the sides, and is open to CO 2 at the top. Our primary objective is to understand effects of geochemical reactions of different rates and orders on density driven natural convection of CO 2 due to concentration and geothermal gradients. We present propagation of CO 2 plumes over long period of time by analyzing different c: Damkohler number (0.01 ≤ Da ≥ 10 5 ) with variation of 1st, 2nd, and 3rd order reactions; solutal Rayleigh number (500 ≤2000); the buoyancy ratio (2 ≤ N 100); Dykstra-Parsons coefficient (0.35 ≤ V dp 0.85); and fixed Lewis and Prandtl numbers. Reaction order is set from relevant stoichiometric ratio. In each case results are quantified in term of percentage of CO 2 deposition. It is found that mineral interactions make traceable difference of depositions as reaction rate increases, especially when Da > 103. In one hand, compared to very minimum geochemical interaction ( Da ∼ 0.01) the strong reaction effects ( Da ∼ 104) can make difference of more than 5% in the period of 500 years of trapping. On the other hand, at a fixed equivalent Da reaction order also makes substantial distinction as deposition time passes. Heterogeneity plays a vital role. Geothermal gradient has very negligible effect, however only after very long time (> 500 years).

Linear and Nonlinear Stability Analysis of Double Diffusive Convection in a Maxwell Fluid Saturated Porous Layer with Internal Heat Source

The onset of double diffusive convection is investigated in a Maxwell fluid saturated porous layer with internal heat source. The modified Darcy law for the Maxwell fluid is used to model the momentum equation of the system, and the criterion for the onset of the convection is established through the linear and nonlinear stability analyses. The linear analysis is obtained using the normal mode technique, and the nonlinear analysis of the system is studied with the help of truncated representation of Fourier series. The effects of internal Rayleigh number, stress relaxation parameter, normalized porosity, Lewis number, Vadasz number and solute Rayleigh number on the stationary, and oscillatory and weak nonlinear convection of the system are shown numerically and graphically. The effects of various parameters on transient heat and mass transfer are also discussed and presented analytically and graphically.

On the onset of thermosolutal instability in a layer of an elastico-viscous nanofluid in porous medium

The thermosolutal instability problem in a horizontal layer of an elasticoviscous nanofluid in porous medium is considered. Walters' (model B') fluid model is employed to describe the rheological behavior of the nanofluid and for the porous medium, the Darcy model is employed. From the linear stability analysis based upon normal modes analysis method, the dispersion relation accounting for the effect of various parameters is derived. The onset criterion for stationary convection is analytically derived and it is observed that the Walters' (model B') elastico-viscous nanofluid fluid behaves like an ordinary Newtonian nanofluid. The effects of solutal Rayleigh Number, thermo-nanofluid Lewis number, thermosolutal Lewis number, Soret and Dufour parameter on the stability of stationary convection are investigated.

Double‐Diffusive Convective Transport in a Nanofluid‐Saturated Porous Layer with Cross Diffusion and Variation of Viscosity and Conductivity

AbstractThe onset of double‐diffusive nanofluid convection in a fluid‐saturated horizontal porous layer is studied with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. The Darcy model has been used for the porous medium, while the nanofluid incorporates the effects of Brownian motion along with thermophoresis. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation in the vertical direction of conductivity and viscosity. In addition, the thermal energy equation includes regular diffusion and cross diffusion terms. The linear stability analysis is based on the normal mode technique, while for nonlinear analysis, minimal representation of the truncated Fourier series representation involving only two terms has been used. It is found that for the stationary mode the Soret parameter, Dufour parameter, viscosity ratio, and conductivity ratio have a stabilizing effect, while the solutal Rayleigh number destabilizes the system. For the oscillatory mode, the Soret parameter, Dufour parameter, and viscosity ratio have a stabilizing effect while the solutal Rayleigh number and conductivity ratio destabilize the system. For steady finite amplitude motions, the heat and mass transport decreases with an increase in the values of the Dufour parameter and solutal Rayleigh number. The Soret parameter enhances the solute concentration Nusselt number while it retards the thermal Nusselt number and concentration Nusselt number. The viscosity ratio and conductivity ratio enhances the heat and mass transports. We also study the effect of time on transient Nusselt numbers which is found to be oscillatory when time is small. However, when time becomes very large, all three transient Nusselt values approach a steady value. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 628–652, 2014; Published online 11 November 2013 in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21102

Onset of triply diffusive convection in a Maxwell fluid saturated porous layer

The linear stability of triply diffusive convection in a binary Maxwell fluid saturated porous layer is investigated. Applying the normal mode method theory, the criterion for the onset of stationary and oscillatory convection is obtained. The modified Darcy–Maxwell model is used as the analysis model, this allows us to make a thorough investigation of the processes of viscoelasticity and diffusions that causes the convection to set in through oscillatory rather than stationary. The effects of Vadasz number, normalized porosity parameter, relaxation parameter, Lewis number and solute Rayleigh number on the system are presented numerically and graphically.

Linear and weakly nonlinear triple diffusive convection in a couple stress fluid layer

The effect of couple stresses on linear and weakly nonlinear stability of a triply diffusive fluid layer is investigated. Several departures not observed either in singly or doubly diffusive couple stress fluid layer have been identified while analyzing the linear stability of the problem. In contrast to the doubly diffusive couple stress fluid system, oscillatory convection is found to occur even if the diffusivity ratios are greater than unity. The presence of couple stress is to increase the threshold value of solute Rayleigh number beyond which oscillatory convection is preferred. Moreover, disconnected closed oscillatory neutral curves are identified for certain choices of physical parameters indicating the requirement of three critical values of Rayleigh number to specify the linear stability criteria instead of the usual single value. Besides, heart-shaped oscillatory neutral curves are also found to occur in some cases and the effect of couple stress parameter on some of these unusual behaviors is analyzed. A weakly nonlinear stability analysis is performed using modified perturbation technique and the stability of steady bifurcating non-trivial equilibrium solution is discussed. Heat and mass transfer are calculated in terms of Nusselt numbers and the influence of various physical parameters on the same is discussed in detail.

Onset of Triply Diffusive Convection in a Maxwell Fluid Saturated Porous Layer

The linear stability of triply diffusive convection in a binary Maxwell fluid saturated porous layer is investigated. Applying normal mode analysis , the criterion for the onset of stationary and oscillatory convection is obtained. The modified Darcy-Maxwell model is used as the analysis model. This allows us to make a thorough investigation of the processes of viscoelasticity and diffusions that causes the convection to set in through oscillatory rather than stationary. The effects of the parameters of Vadasz number, normalized porosity parameter, relaxation parameter, Lewis number and solute Rayleigh number are presented.

Effect of Vertical Vibrations on the Onset of Binary Convection

In the present work the linear stability analysis of double diffusive convection in a binary fluid layer is performed. The major intention of this study is to investigate the influence of time-periodic vertical vibrations on the onset threshold. A regular perturbation method is used to compute the critical Rayleigh number and wave number. A closed form expression for the shift in the critical Rayleigh number is calculated as a function of frequency of modulation, the solute Rayleigh number, Lewis number, and Prandtl number. These parameters are found to have a significant influence on the onset criterion; therefore the effective control of convection is achieved by proper tuning of these parameters. Vertical vibrations are found to enhance the stability of a binary fluid layer heated and salted from below. The results of this study are useful in the areas of crystal growth in micro-gravity conditions and also in material processing industries where vertical vibrations are involved

Numerical investigation of double diffusive natural convection of CO2 in a brine saturated geothermal reservoir

In geologic sequestration or in CO2-based geothermal systems, CO2 is present on top of the brine phase. In this study we performed a numerical analysis of a geothermal reservoir that is impermeable from the sides and is open to CO2 at the top. For this configuration, double diffusive natural convection due to density and temperature differences across the height enhance the mass transfer rate of CO2 into the initially stagnant brine. The analysis is done using mass, momentum, energy conservation laws, and the Darcy laws. The objective is to understand the diffusion of CO2 over long periods of time after sequestration into a subsurface porous media geothermal aquifer. The problem parameters are the solutal Rayleigh number (100≤Ras≤10,000), the buoyancy ratio (2≤N≤100), the cavity aspect ratio (0.5≤A≤2), and a fixed Lewis number (Le=301). Numerical computations do not exhibit natural convection effects for homogeneous initial conditions. Hence a sinusoidal perturbation is added for the initial top boundary condition. It is found that the CO2 plumes move faster when Ras is increased, however they slow down with decreasing N. For every simulation run, the average CO2 concentration (S¯=(∑ini∑jnjci,j/ni×nj)) is computed. Higher concentration rates in laterally wide reservoirs make better candidates than deeper aquifers for CO2 sequestration.

Onset of Darcy‐Brinkman Convection in a Binary Viscoelastic Fluid‐Saturated Porous Layer with Internal Heat Source

The onset of Darcy‐Brinkman convection in a binary viscoelastic fluid‐saturated sparsely packed porous layer with an internal heat source is studied using both linear and nonlinear stability analyses. The Oldroyd‐B model is employed to describe the rheological behavior of binary fluid. An extended form of the Darcy‐Oldroyd law incorporating Brinkman's correction and time derivative is used to describe the flow through a porous layer. The onset criterion for stationary, oscillatory, and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion, and viscoelasticity that causes the convection to set in through an oscillatory mode rather than a stationary mode. The effect of internal Rayleigh number, relaxation and retardation parameters, solute Rayleigh number, Darcy number, Darcy‐Prandtl number, and Lewis number on the stability of a system is investigated and is shown graphically. The nonlinear theory based on the truncated representation of the Fourier series method is used to find heat and mass transfer. The transient behavior of the Nusselt and Sherwood numbers is obtained using numerical methods. Some known results are recovered for the particular cases of the present study. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 42(8): 676–703, 2013; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21056

Double diffusive convection of a micropolar fluid saturated in a sparsely packed porous medium

AbstractThis study examines the double diffusive convection of a sparsely packed micropolar fluid‐saturated porous medium by using a linear stability analysis. The Darcy–Brinkman–Forchheimer model is employed for the porous fluid layer. The stability criterion is sought analytically with the simple free‐free, iso‐thermal, and iso‐solutal boundary conditions. The dependence of stationary or oscillatory convection on the porous parameter, Lewis number, solutal Rayleigh number, and parameters involved in micropolar fluids is drawn and discussed. The results show that the critical wave number is found to be insensitive to the variation of governing parameters except for the porous parameter. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21052

Onset of Double-Diffusive Reaction–Convection in an Anisotropic Rotating Porous Layer

The linear and non-linear stability of a rotating double-diffusive reaction–convection in a horizontal anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated considering a Darcy model that includes the Coriolis term. The effect of Taylor number, mechanical, and thermal anisotropy parameters, reaction rate, solute Rayleigh number, Lewis number, and normalized porosity on the stability of the system is investigated. We find that the Taylor number has a stabilizing effect, chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite-amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.

Unsteady Double Diffusive Natural Convection in Porous Media-Application to CO2 Storage in Deep Saline Aquifer Reservoirs

Storage capacity of carbon dioxide (CO2) in deep saline aquifers is a challenging task. However, an assessment must be performed to determine whether there is sufficient capacity in a storage site for any CO2 sequestration project. We evaluated the CO2 storage capacity for a simplified reservoir system, which the layered potential storage formations are overlaid by sealing cap rock.In this paper, we aim to investigate numerically the storage of CO2 at supercritical conditions in deep saline aquifer reservoirs. Numerical simulations were carried out on non-deformable saturated porous material inside a vertical enclosure, assumed to be impermeable and isolated on three sides. The porous medium is considered to be homogeneous and isotropic with constant thermo-physical properties with the exception of the density of the fluid varying according to Boussinesq approximations.A dynamic model assuming that flow is two-dimensional and obeying to Darcy's law for motion has been used. We assume also that the thermal equilibrium assumption is valid. The set of the conservation equations along with the appropriate initial and boundary conditions have been resolved numerically by the classical finite volume method [1]. Spatio-temporal variations of different state variables such as pressure, velocity, temperature and concentration were numerically simulated and plotted versus different controlling parameters particularly, thermal and solutal Rayleigh numbers and Lewis number. Great attention was paid to examine the sensitivity of heat and mass transfer rates according to the reservoir form and the operating conditions.