Modified Conductivity Ratio Research Articles

Onset of thermovibrational filtration convection: Departure from thermal equilibrium

A theoretical investigation is made to understand the onset of thermovibrational convection in a fluid saturated horizontal porous layer subjected to isothermal heating either at the bottom or at the top. Attention is paid to the situation in which the solid and fluid phases of the porous medium may fail to obey thermal equilibrium locally. Vertical harmonic vibrations of arbitrary amplitude and frequency are considered. The threshold for dynamic instability is found via synchronous and subharmonic resonant modes exploiting the Floquet theory. The nonequilibrium effect is felt only for intermediate values of the interphase heat transfer coefficient H. It is found that H restrains the onset of convection whereas γ, the porosity modified conductivity ratio, encourages it. γ constricts the convective cells ensuing at the threshold except when the layer heated from below is undergoing small amplitude vibrations. Small values of γ expose the competition between synchronous and subharmonic modes for a wider range of vibrational frequencies.

Onset of Darcy–Brinkman Ferroconvection in a Rotating Porous Layer Using a Thermal Non-Equilibrium Model: A Nonlinear Stability Analysis

This article presents a nonlinear stability analysis of a rotating thermoconvective magnetized ferrofluid layer confined between stress-free boundaries using a thermal non-equilibrium model by the energy method. The effect of interface heat transfer coefficient \({( {{\mathcal H}^{\prime}})}\), magnetic parameter (M3), Darcy–Brinkman number \({( {\hat{{\rm D}}{\rm a}})}\), and porosity modified conductivity ratio (γ′) on the onset of convection in the presence of rotation \({({T_{{\rm A}_1}})}\) have been analyzed. The critical Rayleigh numbers predicted by energy method are smaller than those calculated by linear stability analysis and thus indicate the possibility of existence of subcritical instability region for ferrofluids. However, for non-ferrofluids stability and instability boundaries coincide. Asymptotic analysis for both small and large values of interface heat transfer coefficient \({({{\mathcal H}^{\prime}})}\) is also presented. A good agreement is found between the exact solutions and asymptotic solutions.

Numerical Simulation of Natural Convection in Wavy Porous Cavities Under the Influence of Thermal Radiation Using a Thermal Non-equilibrium Model

Continuum equations governing thermal non-equilibrium modeling of steady natural convection inside wavy enclosures with the effect of thermal radiation are developed. These equations account for such effects as the inter-phase heat transfer coefficient effect, the thermal radiation effect, the modified conductivity ratio effect and the Rayleigh number effect. Finite difference method is employed to solve these equations and comparisons between previous published works are presented. Numerical results for the flow and heat transfer for the fluid and solid phases are obtained for various combinations of the physical parameters. Graphical and tabular results illustrating interesting features of the physics of the problem are presented and discussed.

Effect of rotation on the onset of thermal convection in a sparsely packed porous layer using a thermal non-equilibrium model

Linear and nonlinear stability of a rotating fluid-saturated sparsely packed porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium. The extended Darcy–Brinkman model that includes the time derivative and Coriolis terms is employed as a momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for both stationary and oscillatory convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of rotation and thermal diffusion that causes the convection to set in through oscillatory mode rather than stationary. The rotation inhibits the onset of convection in both stationary and oscillatory mode. The Darcy number stabilizes the system towards the oscillatory mode, while it has dual effect on stationary convection. Besides, the effect of porosity modified conductivity ratio, Darcy–Prandtl number and the ratio of diffusivities on the stability of the system is investigated. The nonlinear theory is based on the truncated representation of Fourier series method. The effect of thermal non-equilibrium on heat transfer is brought out. The transient behavior of the Nusselt number is investigated by using the Runge–Kutta method. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.

Stationary Fingering Instability in a Non-equilibrium Porous Medium with Coupled Molecular Diffusion

Finger type double diffusive convective instability in a fluid-saturated porous medium is studied in the presence of coupled heat-solute diffusion. A local thermal non-equilibrium (LTNE) condition is invoked to model the Darcian porous medium which takes into account the energy transfer between the fluid and solid phases. Linear stability theory is implemented to compute the critical thermal Rayleigh number and the corresponding wavenumber exactly for the onset of stationary convection. The effects of Soret and Dufour cross-diffusion parameters, inter-phase heat transfer coefficient and porosity modified conductivity ratio on the instability of the system are investigated. The analysis shows that positive Soret mass flux triggers instability and positive Dufour energy flux enhances stability whereas their combined influence depends on the product of solutal Rayleigh number and Lewis number. It also reveals that cell width at the convection threshold gets affected only in the presence of both the cross-diffusion fluxes. Besides, asymptotic solutions for both small and large values of the inter-phase heat transfer coefficient and porosity modified conductivity ratio are found. An excellent agreement is found between the exact and asymptotic solutions.

Linear and nonlinear double diffusive convection in a rotating sparsely packed porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and nonlinear stability analyses. The Brinkman model that includes the Coriolis term is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for the energy equation. The onset criterion for stationary, oscillatory, and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal diffusion, solute diffusion, and rotation that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number, and Taylor number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.

The onset of double diffusive convection in a sparsely packed porous layer using a thermal non-equilibrium model

We examine the effect of local thermal non-equilibrium on double diffusive convection in a fluid-saturated sparsely packed porous layer heated from below and cooled from above, using both linear and nonlinear stability analyses. The Brinkman model is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for the energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that a small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusion that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, ratio of diffusivities, Vadasz number and Darcy number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.

Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.

Double diffusive convection in a porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and nonlinear stability analyses. The Darcy model with time derivative term is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusion that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, ratio of diffusivities and Vadasz number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.

Thermal convection in a rotating porous layer using a thermal nonequilibrium model

Linear stability of a rotating fluid-saturated porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium. The extended Darcy model, which includes the time derivative and Coriolis terms, is employed as a momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for both stationary and oscillatory convection is derived analytically. It is found that a small interphase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of rotation and thermal diffusion that causes the convection to set in through the oscillatory mode rather than the stationary one. The rotation inhibits the onset of convection in both stationary and oscillatory mode. In addition, the effect of porosity modified conductivity ratio, Darcy-Prandtl number, and the ratio of diffusivities on the stability of the system is investigated. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the Nusselt number. The effect of thermal nonequilibrium on heat transfer is brought out. The transient behavior of the Nusselt number is also investigated by solving the finite amplitude equations using Runge-Kutta method.

Thermal non-equilibrium modeling of heat transfer through vertical annulus embedded with porous medium

Present study is undertaken to investigate the steady state heat transfer in a porous medium fixed in a vertical annular cylinder. A two-temperature, thermal non-equilibrium model is used to analyse the heat transfer behavior in the porous medium. The Darcy model of flow is employed. The inner surface of the vertical annulus is maintained at constant wall temperature and the outer surface remains at ambient temperature. The heat transfer is assumed to take place by natural convection and radiation. The equations governing the flow behavior are non-dimensionalised using suitable non-dimensional parameters and then solved using finite element method. Three-node triangular element is used to mesh the domain. The effect of various parameters such as aspect ratio, radius ratio of annulus, modified conductivity ratio, inter-phase heat transfer coefficient, radiation parameter and Rayleigh number on the heat transfer behavior is discussed.

Numerical analysis of convection conduction and radiation using a non-equilibrium model in a square porous cavity

Numerical analysis of heat transfer by convection, conduction and radiation in a saturated porous medium enclosed in a square cavity is investigated using a thermal non-equilibrium model. The flow is assumed to follow Darcy law. The governing partial differential equations are non-dimensionalised and solved numerically using finite element method. The left vertical surface of the square cavity is maintained at an isothermal temperature T h and the right vertical surface at T c such that T c < T h . The top and bottom surfaces of the cavity are assumed to be adiabatic. Results are presented in terms of Nusselt number for fluid, Nusselt number for solid and total Nusselt number, for various parameters such as inter-phase heat transfer coefficient, modified conductivity ratio, radiation parameter and Rayleigh number.

Thermal response due to continuous cooling in a thermomechanical treatment process

A mathematical model for the thermomechanical treatment process is developed when a hot rolled product in the form of a plate passes through a quiescent quenching bath. The coupled heat conduction equation, that accounts for the movement of the hot plate, and the heat convection equation for the bath are solved using the integral method. Closed-form solutions associated with design parameters of this process, like the evolution of temperature in the plate due to its continuous cooling, the rate of heat transfer and the temperature build-up in the bath are obtained. They are controlled by independent parameters—Brinkman number, Br, dimensionless modified conductivity ratio, N, and diffusivity ratio, M. The temperature profile developed in the plate and the associated heat transfer exhibit that for any value of Br, the low values of N and M cause a significant change in the temperature of the plate, resembling the quenching process. With the change of N and M to high values, the temperature within the plate becomes almost uniform. The plate becomes normalized. Increasing Br reduces the rate of heat transfer but it has an insignificant effect upon the temperature change. To check the validity of the present mathematical model it is reduced to a problem investigated previously. A close agreement is observed.