The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

A theoretical analysis of forced convection in a porous-saturated circular tube: Brinkman–Forchheimer model

Fully developed forced convection inside a circular tube filled with saturated porous medium and with uniform heat flux at the wall is investigated on the basis of a Brinkman–Forchheimer model. The matched asymptotic expansion method is applied at small Darcy numbers. For large Darcy numbers, the solution for the Brinkman–Forchheimer momentum equation is found in terms of an asymptotic expansion. Once the velocity distribution is determined, the energy equation is solved using the same asymptotic technique. The results for the two limiting cases of clear fluid and Darcy flow conditions show good agreement with those available in the literature.     

Fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media

We review and discuss diffusion and hydrodynamic dispersion in a heterogeneous porous medium. Two types of heterogeneities are considered. One is percolation disorder in which a fraction of the pores do not allow transport to take place at all. In the other type, the permeabilities of various regions of the pore space are fractally distributed with long-range correlations. Both systems give rise to unusual transport in which the mean square displacement <r ²(t)> of a particle grows nonlinearly with time. Depending on the heterogeneities and the mechanism of diffusion and disperison, we may havefractal transport in which <r ²> growsslower than linearly with time, orsuperdiffusive transport in which <r ²> growsfaster than linearly with time. We show that percolation models can give rise to both types of transport with scale-dependent transport coefficients such as diffusivity and dispersion coefficients, which are consistent with many experimental observations.     

A LGA model for dispersion in heterogeneous porous media

The lattice gas automaton (LGA) model proposed in the previous paper is applied to the problem of simulating dispersion and mixing in heterogeneous porous media. We demonstrate here that tracer breakthrough profiles and longitudinal dispersion coefficients can be computed for heterogeneous porous media.     

Stability analysis of thermosolutal convection in a horizontal porous layer using a thermal non-equilibrium model

A stability analysis is carried out to investigate the onset of thermosolutal convection in a horizontal porous layer when the solid and fluid phases are not in a local thermal equilibrium, and the solubility of the dissolved component depends on temperature. To study how the reaction and thermal non-equilibrium affect the double-diffusive convection, the effects of scaled inter-phase heat transfer coefficient H and dimensionless reaction rate k on thermosolutal convection are discussed . The critical Rayleigh number and the corresponding wave number for the stability and overstability convections are obtained. Specially, asymptotic analysis for both small and large values of H and k is presented, and the corresponding asymptotic solutions are compared with numerical results. At last, a nonlinear stability analysis is presented to study how H and k affect the Nusselt number.     

Bifurcation analysis for thermal convection in a rotating porous layer

The linear and weakly nonlinear thermal convection in a rotating porous layer is investigated by constructing a simplified model involving a system of fifth-order nonlinear ordinary differential equations. The flow in the porous medium is described by Lap wood-Brinkman-extended Darcy model with fluid viscosity different from effective viscosity. Conditions for the occurrence of possible bifurcations are obtained. It is established that Hopf bifurcation is possible only at a lower value of the Rayleigh number than that of simple bifurcation. In contrast to the non-rotating case, it is found that the ratio of viscosities as well as the Darcy number plays a dual role on the steady onset and some important observations are made on the stability characteristics of the system. The results obtained from weakly nonlinear theory reveal that, the steady bifurcating solution may be either sub-critical or supercritical depending on the choice of physical parameters. Heat transfer is calculated in terms of Nusselt number.     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail.     

The Starting Flow in Ducts Filled with a Darcy–Brinkman Medium

Analytical solutions are found for the transient starting flow due to a sudden pressure gradient in cylindrical, rectangular, and parallel plate ducts fill with a Darcy– Brinkman porous medium. It is found that, for all geometries, the initial velocity front is flat. It eventually becomes more parabolic for small porous media parameter s but remains flat for large s. The boundary layer thickness is of order (1/s). The transient is also shorter (proportional to exp(−s ² t)) for large s.     

Thermal-hydraulic core analysis of the VVER-1000 reactor using a porous media approach

In this study, a thermal-hydraulic analysis of the VVER-1000 reactor core is performed using a porous media approach. Based on this approach, each fuel assembly was modeled and was divided into a network of lumped regions, each of which was characterized by a volume average parameter. The conservation equations of mass, linear momentum and energy are derived and discretized using the finite volume method in a hexagonal coordinate system. The pressure, velocity and temperature fields are achieved using a numerical analysis of the above mentioned coupled equations. To validate the applied approach, the numerical analysis and COBRA EN code results were compared and showed good agreement.     

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.     

Response of saturated porous media subjected to local thermal loading on the surface of semi-infinite space

Heat source function method is adopted in the present paper to derive elementary solutions of coupled thermo-hydro-mechanical consolidation for saturated porous media under conjunct actions of instantaneous point heat source, instantaneous point fluid source and constant volume force. By using the so-called fictitious heat source method and images method, the solutions of a semi-infinite saturated porous medium subjected to a local heat source with time-varied intensity on its free surface are developed from elementary solutions. The numerical integral methods for calculating the unsteady temperature, pore pressure and displacement fields are given. The thermomechanical response are analyzed for the case of a circular planar heat source. Besides, the thermal consolidation characteristics of a saturated porous medium subjected to a harmonic thermal loading are also given, and the fluctuation processes of the field variables located below the center of heat source are analyzed. The project supported by the National Natural Science Foundation of China (50578008) The English text was polished byYunming Chen.     

The effect of local thermal non-equilibrium on conduction in porous channels with a uniform heat source

We examine the effect of local thermal non-equilibrium on the steady state heat conduction in a porous layer in the presence of internal heat generation. A uniform source of heat is present in either the fluid or the solid phase. A two-temperature model is assumed and analytical solutions are presented for the resulting steady-state temperature profiles in a uniform porous slab. Attention is then focussed on deriving simple conditions which guarantee local thermal equilibrium.     

Effects of thermal dispersion and stratification on combined convection on a vertical surface embedded in a porous medium

The effects of thermal dispersion and thermal stratification on mixed convection about a vertical surface in a porous medium are studied. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations are solved on the basis of the local similarity approximation. The results indicate that both dispersion and stratification effects have considerable influence on the heat transfer rate.     

Check on a nonlocal Maxwellian theory of sound propagation in fluid-saturated rigid-framed porous media

A macroscopic nonlocal theory of sound propagation in homogeneous rigid-framed porous media permeated with a viscothermal fluid has been recently proposed in this journal. It accounts for the first time for the full temporal and spatial dispersion effects, independently of the nature of the microgeometry. In this paper this new Maxwellian theory is validated in the case of sound propagation in cylindrical circular tubes, by showing that it matches exactly the long-known direct Kirchhoff–Langevin’s solutions.     

Experimental investigation of characteristic length scale in periodic heterogeneous porous media

An experimental investigation of scale-dependent dispersion in periodic heterogeneous porous media was conducted. Models with two-, three- and four-layer periodic heterogeneities were constructed to investigate the effect of heterogeneity size on the scale-dependence of dispersion. Longitudinal dispersion coefficients were determined as a function of column length by measuring the breakthrough of a continuous injection of potassium chloride tracer solution. Chloride ion concentration was monitored by recording the millivolt potential of silver/silver chloride electrodes placed at intervals along the length of the column. In all three models, dispersion appeared to be scale dependent up to a distance of approximately 20–30 times the size of the repeated heterogeneity group (hydraulic unit). Because all three models suggested a similar dependence, it was concluded that a medium with periodic heterogeneity may likely be characterized by the scale of its hydraulic unit.     

Influence of mixture sensitivity and pore size on detonation velocities in porous media

Experiments have been carried out to determine the dependence of the detonation velocity in porous media, on mixture sensitivity and pore size. A detonation is established at the top end of a vertical tube and allowed to propagate to the bottom section housing the porous bed, comprised of alumina spheres of equal diameter (1–32 mm). Several of the common detonable fuels were tested at atmospheric initial pressure. Results indicate the existence of a continuous range of velocities with change in Φ, spanning the lean and the rich propagation limits. For all fuels in a given porous bed, the velocity decreases from a maximum value at the most sensitive mixture near Φ≈1 (minimum induction length), toV/V _CJ≈0.3 at the limits. A decrease in pore size brings about a reduction inV/V _CJ and a narrowing of the detonability range for each fuel. For porous media comprised of spherical particles, it was possible to correlate the velocity data corresponding to a variety of different mixtures and for a broad range of particle sizes, using the following empirical expression:V/V _CJ=[1–0.35 log(d _c /d _p)]±0.1. The critical tube diameterd _c is used as a measure of mixture sensitivity andd _p denotes the pore diameter. An examination of the phenomenon at the composition limits, suggests that wave failure is controlled by a turbulent quenching mechanism.     

Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a porous medium

Stability analysis of double-diffusive convection for viscoelastic fluid with Soret effect in a porous medium is investigated using a modified-Maxwell-Darcy model. We use the linear stability analysis to investigate how the Soret parameter and the relaxation time of viscoelastic fluid effect the onset of convection and the selection of an unstable wavenumber. It is found that the Soret effect is to destabilize the system for oscillatory convection. The relaxation time also enhances the instability of the system. The effects of Soret coefficient and relaxation time on the heat transfer rate in a porous medium are studied using the nonlinear stability analysis, the variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Some previous results can be reduced as the special cases of the present paper.