Equations Brinkman

As mentioned earlier, the expression forfd is obtained under conditions of no inertia. If we further assume the resin is Newtonian (i.e., r = p[V Ur + V / ])) and the fiber phase is stationary, then Equation 5.25 can be simplified to the well-known Brinkman equation [22],... 

Moreover, if one assumes that the (Ur) changes very slowly on the length scale of the porous media, (i.e., ), then the viscous stress term in the Brinkman equation can be neglected and this equation reduces to ... 

Concerning the flow in the porous media, the use of the Brinkman equation allows to set the continuity in the velocity profiles in the channel and in the porous medium (Nield and Bejan, 1999). 

In fixed bed chemical reactor analysis it is common to assume uniform flow distribution within the bed. The reality however is different. Due to a change of the average porosity near the wall [ 1, 2,3],(Figure 1.) -e=1 at the wall - the flow velocity increases until close to the wall and is reduced again because of the non slip condition (Figure 2.) The artificial flow profile is described by the Brinkman equation... 

To obtain the penetration depth quantitatively, Milner assumed that the brush profile was undistorted by the flow and invoked the Brinkman equation [56] for flow in a porous media,... 

This idea that the solvent flow field can be approximated by the Brinkman equation has been used in several recent simulations of a polymer brush in simple shear flow. In these simulations, the solvent is not included explicitly but it s effect is modeled using the Brinkman equation. Lai and Binder [65] and Lai and Lai [66], using a bond fluctuation lattice model, and Miao et al. [67], using a continuum model, studied the properties of a dense polymer brush in a flow field by modifying the standard Metropolis Monte Carlo transition probability to take into account the effective force acting upon the brush chains by the moving sol-... 

The hierarchy of equations thereby obtained can be closed by truncating the system at some arbitrary level of approximation. The results eventually obtained by various authors depend on the implicit or explicit hypotheses made in effecting this closure—a clearly unsatisfactory state of affairs. Most contributions in this context aim at calculating the permeability (or, equivalently, the drag) of a porous medium composed of a random array of spheres. The earliest contribution here is due to Brinkman (1947), who empirically added a Darcy term to the Stokes equation in an attempt to represent the hydrodynamic effects of the porous medium. The so-called Brinkman equation thereby obtained was used to calculate the drag exerted on one sphere of the array, as if it were embedded in the porous medium continuum. Tam (1969) considered the same problem, treating the particles as point forces he further assumed, in essence, that the RHS of Eq. (5.2a) was proportional to the average velocity and hence was of the explicit form... 

The Brinkman equation (1) transitions properly from Stokes flow (in the absence of inertia) to Darcy flow in a porous medium ... 

L. Durlofsky and J. F. Brady, Analysis of the Brinkman equation as a model for flow in porous... 

Problem 12-18. Buoyancy-Driven Instability of a Fluid Layer in a Porous Medium Based on the Darcy-Brinkman Equations. A more complete model for the motion of a fluid in a porous medium is provided by the so-called Darcy Brinkman equations. In the following, we reexamine the conditions for buoyancy-driven instability when the fluid layer is heated from below. We assume that inertia effects can be neglected (this has no effect on the stability analysis as one can see by reexamining the analysis in Section H) and that the Boussinesq approximation is valid so that fluid and solid properties are assumed to be constant except for the density of the fluid. The Darcy Brinkman equations can be written in the form... 

Note that the viscosity used in both the microscopic and macroscopic viscous terms is the fluid viscosity. Lundgren [26], in giving justification to the Brinkman equation, shows that for spherical particles p = pg(e), where 0 < g(e) and g(l) = 1 but others have used p > p. 

Unlike conventionally used volume-averaged equations, equation 20 does reduce to the Brinkman equation 5 when the inertial effects are negligible and to Darcy s law, equation 3, when both inertial and multidimensional effects are negligible. 

Extensions of Brinkman s formalism [140, 141] have facilitated the numerical solving of the Navier-Stokes equations for the flow outside the aggregate under laminar flow conditions, and the application of the Brinkman equation for flow inside... 

For forward osmosis or pressure-retarded osmosis, since the ICP effects occurring inside the PSL of permeable membrane are also investigated, an alternative momentum governing equation for porous media, Brinkman equation, should be applied for porous region of the membrane [8] ... 

Other alternative to the Darcy model is the Brinkman equation. Omitting the inertial terms, it has the form... 

Conservation Equations The computational model consists of steady-state continuity and momentum equations to simulate flow through the non-catalytic part of the reactor and the continuity and momentum equations, known as the Brinkman equations, to simulate flow through the catalytic bed of the reactor. 

The flow in the gas channels and in the porous gas diffusion electrodes is described by the equations for the conservation of momentum and conservation of mass in the gas phase. The solution of these equations results in the velocity and pressure fields in the cell. The Navier-Stokes equations are mostly used for the gas channels while Darcy s law may be used for the gas flow in the GDL, the microporous layer (MPL), and the catalyst layer [147]. Darcy s law describes the flow where the pressure gradient is the major driving force and where it is mostly influenced by the frictional resistance within the pores [145]. Alternatively, the Brinkman equations can be used to compute the fluid velocity and pressure field in porous media. It extends the Darcy law to describe the momentum transport by viscous shear, similar to the Navier-Stokes equations. The velocity and pressure fields are continuous across the interface of the channels and the porous domains. In the presence of a liquid phase in the pore electrolyte, two-phase flow models may be used to account for the interaction between the gas phase and the liquid phase in the pores. When calculating the fluid flow through the inlet and outlet feeders of a large fuel cell stack, the Reynolds-averaged Navier-Stokes (RANS), k-o), or k-e turbulence model equations should be used due to the presence of turbulence. 

A number of different approaches are proposed and used in modeling flow through porous media. Some of the most popular approaches include (i) Darcy s law, (ii) Brinkman equation, and (iii) a modified Navier-Stokes equation. In the absence of the bulk fluid motion or advection transport, the reaction gas species can only transport through the GDL and CL by the diffusion mechanisms, which we will discuss in a later section. 

Martys, N. Improved approximation of the Brinkman equation using lattice Boltzmarm method. Physics of Fluids 13(6) 1807-1810, 2001. 

Axial flow in the brush layer can be determined from the Debye-Brinkman equations [45, 80] ... 

To account for the viscous drag along bounding walls, the Brinkman equation is often used. The Brinkman equation describes the flow in porous media in cases where the transport of momentum by shear stresses in the fluid is not ignored. The model extends to include a term that accounts for the viscous transport in the momentum balance and introduces the velocities in the spatial directions as dependent variables. 

The term (-ur]lk) in Eq. 3.23 is the Darcy resistance term, and the term (rjW u) is the viscous resistance term the driving force is still considered to be the pressure gradient. When the permeability k is low, the Darcy resistance dominates the Navier-Stokes resistance, andEq. 3.23 reduces to Darcy s law. Therefore, the Brinkman equation has the advantage of considering both viscous drag along the walls and Darcy effects within the porous medium itself. In addition, because Brinkman s equation has second-order derivatives of u, it can satisfy no-slip conditions at solid surfaces bounding the porous material (e.g. the walls of a packed bed reactor), whereas Darcy s law cannot. In that sense, Brinkman s equation is more exact than Darcy s law. 

To calculate Eq. 4.23, the flow velocity distribution within the package U (u, v, w) must be defined. This involves the fluid motion equation in porous media. As discussed in Chapter 3, Darcy s law and Brinkman equation can be used to characterise the flow behavioirr within the package. 

In order to investigate the influence of shear stresses on the flow property in a yam assembly, the Brinkman equations are used to model flow in a porous medium, as shown in Eqs 4.36 and 4.37. 

It should be noted that in Eq. 4.38 the velocity in both x and y-directions is considered based on the Brinkman equation approach. 

The conditions for botmdary 5, to couple Darcy s law (or the Brinkman equation) and the value of the pressttre, are defined by setting ... 

The flow of dye hquor in an open channel (tube) and in a porous package was coupled here in an attempt to demonstrate the combined features of flow within the system. The flow is described by the Navier-Stokes equation in the free region, and the Darcy s law and Brinkman equations in the porous... 

Figure 6.4 uses the same simulation result as that shown in Fig. 6.2, with the fluid flow velocity profiles, where the flow in the porous package is defined by the Brinkman equation, but at different cross-section lines along the package. The solid line is the same as that in Fig. 6.2, i.e. the middle of the package (Plate I, y=0), the dotted line is near the top of the package (y=0.06) and the dashed line is near the bottom of the package (y=-0.06). 

Darcy s law and by the Brinkman equation, respectively. It can be seen that, in both approaches, the pressure profiles within the free liquor (tube) are quite stable and almost constant under the different inflow rates the pressure profiles within the porous package decrease almost linearly at the different inflow rates. 

Brinkman equations extend Darcy s law to iiKlude a term that accounts for the viscous transport in the momentum balance and introduce the velocities iu the spatial directious as depeudeut variables. This approach is more robust thau using Darcy s law alone, since it can be applied in a wider range of flow rates and porous package permeabihties. 

The flow velocity distributions within the porous yam assembly are dominated by the pressme rather than the flow velocity distributions within the free liquor circulating in the spindle. This result is obtained using Brinkman equations. It further proves the validity of Darcy s law in characterising the flow within the porous medium, since it agrees with Darcy s assumption that the only driving force for flow in a porous medium is the pressure gradient. 

---