Darcy convection

Neither of these forms of Darcy s law is correct where fluid density varies, such as in convecting flows. In this case, the discharge is given by,... 

In eq 51, the first term represents a convection term, and the second comes from a mass flux of water that can be broken down as flow due to capillary phenomena and flow due to interfacial drag between the phases. The velocity of the mixture is basically determined from Darcy s law using the properties of the mixture. The appearance of the mixture velocity is a big difference between this approach and the others, and it could be a reason the permeability is higher for simulations based on the multiphase mixture model. 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

Figure 6 shows in a cylindrical coordinate (radial/thm-plane) how H2 depletes and N2 pressure builds up with time. At the center of the H2-starved region, the N2 pressure becomes higher than 80 kPaabs after 1 s. The radius of this N2 bubble grows to 7 mm after 10 s and covers 80% of the liquid-water-blocked region at 100 s (near steady-state). The H2 pressure drops quickly to zero at the edge of the N2 bubble, within which the cathode potential rises sharply, and the carbon corrosion rate starts to increase, as shown in Fig. 7. Conventional DM has a value of permeability ranging from 1 to 10 Darcy. There is no impact of DM permeability within its realistic range.14 N2 crossover through the membrane results in N2 pressure build-up in the H2-starved anode region. As a result, convective...

Pressure-driven convective flow, the basis of the pore flow model, is most commonly used to describe flow in a capillary or porous medium. The basic equation covering this type of transport is Darcy s law, which can be written as... 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

In laminar flows through porous media, the pressure is proportional to velocity and C2 can be taken as zero. Ignoring convective acceleration and diffusion, the porous media model can be changed into Darcy s Law ... 

If the Darcy assumptions are used then with forced convective flow over a surface in a porous medium, because the velocity is not assumed to be 0 at the surface, there is no velocity change induced by viscosity near the surface and there is therefore no velocity boundary layer in the flow over the surface. There will, however, be a region adjacent to the surface in which heat transfer is important and in which there are significant temperature changes in the direction normal to the surface. Under many circumstances, the normal distance over which such significant temperature changes occur is relatively small, i.e., a thermal boundary layer can be assumed to exist around the surface as shown in Fig. 10.9, the ratio of the boundary layer thickness, 67, to the size of the body as measured by some dimension, L, being small [15],[16]. 

Hence, this analysis indicates that instability will first occur, i.e., convective motions will first occur, when the Darcy-modified Rayleigh number based on the thickness of the layer reaches 39.5. If Ra is greater than this value, convective motion will occur. This value for / amjn is in good agreement with experiment, some experimental results being shown in Fig. 10.35. 

Nakayama, A. and Pop, I., A Unified Similarity Transformation for Free, Forced and Mixed Convection in Darcy and Non-Darcy Porous Media . Int. J. Heat and Mass Transfer, Vol. 34, pp. 357-367, 1991. 

Kumati, M. and Nath. G.. "Non-Darcy Mixed Convection Flow over a Nonisother-mal Cylinder and Sphere Embedded in a Saturated Porous Medium". J. Heat Transfer. Vol. 112, pp.318-521, 1990. 

Lai, F.C. and Kulacki, F.A., "Non-Darcy Mixed Convection Along a Vertical Wall in a Saturated Porous Medium , J Heat Transfer, Vol 113. pp. 252-255. 1991. 

Nakayama, A., A Unified Theory for Non-Darcy Free, Forced, and Mixed Convection Problems Associated with a Horizontal Line Heat Source in a Porous Medium , J. Heat Transfer, Vol. 116, pp. 508-513, 1994. 

Convection (also known as advection) is the vector, which results from the DARCY or the RICHARDS equations. It describes the flow velocity or the flow distance for a certain time t. In general convection has the major influence on... 

Equations (8.10)—(8.12), tensorial ranks and boundary conditions (8.14)-(8.15) notwithstanding, embody a structure similar in format and symbolism to their counterparts for the transport of passive scalars, e.g., the material transport of the scalar probability density P (Brenner, 1980b Brenner and Adler, 1982), at least in the absence of convective transport. As such, by analogy to the case of nonconvective material transport, the effective kinematic viscosity viJkl of the suspension may be obtained by matching the total spatial moments of the probability density Pu to those of an equivalent coarse-grained dyadic probability density P j, valid on the suspension scale, using a scheme (Brenner and Adler, 1982) identical in conception to that used to determine the effective diffusivity for material transport at the Darcy scale from the analogous scalar material probability density P. In particular, the second-order total moment M(2) (sM, ) of the probability density P, defined as... 

Because it is difficult to account for changes in the properties of the reaction medium (e.g., permeability, thermal conductivity, specific heat) due to structural transformations in the combustion wave, the models typically assume that these parameters are constant (Aldushin etai, 1976b Aldushin, 1988). In addition, the gas flow is generally described by Darcy s law. Convective heat transfer due to gas flow is accounted for by an effective thermal conductivity coefficient for the medium, that is, quasihomogeneous approximation. Finally, the reaction conditions typically associated with the SHS process (7 2(XX) K and p<10 MPa) allow the use of ideal gas law as the equation of state. 

Stefan-Maxwell expression of diffusive fluxes and a Darcy expression of convective ones, was frequently employed, especially for the IMR with separate feed of reactants [30,48,49]. Such studies clearly indicated that the use of the Stefan-Maxwell approach to diffusion has to be preferred over a simple Pick one, especially when large pressure differences are imposed across the membrane. 

Differential mass balances across the membrane combine Stefan-Maxwell-type diffusive fluxes, a surface diffusion term, the Darcy expression for convective fluxes, and the reaction terms. 

Kim et al. [22] modeled microchannel heat sinks as porous structures, while stud3ung the forced convective heat transfer through the microchannels. From the analytical solution, the Darcy number and the effective thermal conductivity ratio were identified as variables of engineering importance. 

Microfiltration and ultrafiltration are the two main filtration techniques for which ceramic membranes have been widely used to date. As described in Section 6.2.1.2, MF and UF ceramic membranes exhibit macro- and mesoporous structure, respectively, which result from packing and sintering of ceramic particles. Liquid flow in such porous media is convective in nature and the simplest description of permeation flux, J, is given by the Darcy s equation [20] ... 

As shown in the preceding parts, kinetic parameters cannot be directly calculated when internal heat transfer limits pyrolysis. A model taking into account both kinetic scheme and heat- mass transfers becomes necessary, A one-dimension model has already been implemented and solved. It features a classical set of equations for heat and mass transfers in porous media, i.c. heat transfer through convection, conduction, radiation and mass transfer due to pressure gradient (Darcy s law) and binary diffusion. Different kinetic schemes from e literature arc and will be tested mass-loss as lumped first order reaction, formation of volatiles, tars and char from decomposition of cellulose, hcmicellulose and lignin [26], the Broido-Shafi2adeh model [30] and the one proposed by Di Blasi [31]. None of them can describe the composition of the volatiles and supplementary schemes have to be found. 

However, if convective transport of heat and species mass in porous catalyst pellets have to be taken into account simulating catal3dic reactor processes, either the Maxwell-Stefan mass flux equations (2.394) or dusty gas model for the mass fluxes (2.427) have to be used with a variable pressure driving force expressed in terms of mass fractions (2.426). The reason for this demand is that any viscous flow in the catalyst pores is driven by a pressure gradient induced by the potential non-uniform spatial species composition and temperature evolution created by the chemical reactions. The pressure gradient in porous media is usually related to the consistent viscous gas velocity through a correlation inspired by the Darcy s law [21] (see e.g., [5] [49] [89], p 197) ... 

This is a semiheuristic volume-averaged treatment of the flow field. The experimental observations of Dybbs and Edwards [27] show that the macroscopic viscous shear stress diffusion and the flow development (convection) are significant only over a length scale of i from the vorticity generating boundary and the entrance boundary, respectively. However, Eq. 9.22 predicts these effects to be confined to distances of the order oi Km and KuDN, respectively. We note that Km is smaller than d. Then Eq. 9.22 predicts a macroscopic boundary-layer thickness, which is not only smaller than the representative elementary volume i when i d, but even smaller than the particle size. However, Eq. 9.22 allows estimation of these macroscopic length scales and shows that for most practical cases, the Darcy law (or the Ergun extension) is sufficient. 

The original equations discussed earlier take into account heat transfer by conduction and convection, heat of evaporation and condensation, capillary effect and Darcy s law, but they ignore the Joule-Thomson effect and the temperature effect of adiabatic expansion of formation liquids and gases. 

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