Brinkman’s equation

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

The theoretical equations relating relative fluidity (l/qrci) to dispersion concentration also became widely known. Among them is Brinkman s equation [31], valid for dispersions of particles with a wide particle size distribution ... 

A very important form of such disturbances is caused by the presence of the wall of the tube containing the packed bed. Vortmeyer and Schuster (1983) have used a variational approach to evaluate the steady two-dimensional velocity profiles for isothermal incompressible flow in rectangular and circular packed beds. They used the continuity equation, Brinkman s equation (1947), and a semiempirical expression for the radial porosity profile in the packed bed to compute these profiles. They were able to show that significant preferential wall flow occurs when the ratio of the channel diameter to the particle diameter becomes sufficiently small. Although their study was done for an idealized situation it has laid the foundation for more detailed studies. Here CFD has definitely contributed to the improvements of theoretical prediction of reactor performance. 

In addition to the Reynolds-number limitations discussed above, a shortcoming of Darcy s law is that it does not allow boimdary conditions to be imposed at porous media interfaces (e.g., a no-slip condition at a reactor wall). This problem is addressed by Brinkman s equation, which is a combination of the Stokes equation and Darcy s equation ... 

Brinkman s equation is restricted to higher-porosity materials It is invalid for < 1 and is best used for >... 

Brinkman s equation represents a variant of the effective medium approximation, which does not describe explicitly the generation of non-laminar liquid motion and conversion of the in-plane surface motion into the normal-to-interface liquid motion. These effects result in additional channels of energy dissipation, which are effectively included in the model by introduction of the Darcy-like resistive force. 

When diffusion coefficients for BSA were measured in dextran solutions by holographic interferometry, the BSA diffusion coefficient decreased by less than a factor of 2 as dextran concentration was increased from 0 to 0.08 g/mL [54] the diffusion coefficient was not a function of dextran molecular weight (the range tested was 9,300 to 2 x 10 ). The BSA diffusion coefficient was described very well by using Brinkman s equation to estimate the influence of hydrodynamic screening due to dextran molecules in the solution. The dextran fibers were assumed to have a radius of 1 nm the hydraulic permeability, k, of the dextran solution was estimated from the semi-empirical relationship ... 

This approach—which uses Brinkman s equation, with an appropriate correlation to permit estimation of the hydraulic permeability from the structural characteristics of the medium—provides a straightforward method for estimating the influence of hydrodynamic screening in polymer solutions predicted diffusion coefficients for probes of 3.4 and 10 nm in dextran solutions (Pf = 1 nm) are shown in Figure 4.9. This approach should be valid for cases in which probe diffusion is much more rapid than the movement of fibers in the network, although it appears to work well for BSA diffusion in dextran solutions, even though the dextran molecules diffuse as quickly as the BSA probes [54]. 

Figure 4.13 Estimation of reduced diffusion coefficient by effective medium approximation. Combination of steric and hydrodynamic effects on reduced diffusion coefficient. The solid lines represent hydrodynamic effect for probe radii of 3.4 and 10 A calculated using Brinkman s equation (see Figure 4.9). The dashed lines represent the combined steric and hydrodynamic effect using Equation 4-40 for the steric effect.

Perturbation theory cannot be applied to describe the effect of the strong roughness. An approach based on Brinkman s equation has been used instead to describe the hydrodynamics in the interfacial region [82]. The flow of a liquid through a nonuniform surface layer has been treated as the flow of a liquid through a porous medium [83-85]. The morphology of the interfacial layer of thickness, L, has been characterized by a local permeability, that depends on the effective porosity of the layer, (j). A number of equations for the permeability have been suggested. For instance, the empirical Kozeny-Carman equation [83] yields a relationship... 

Assumption of Stokes flow to describe the resin flow between the tows and Brinkman s equation to describe the flow inside the tows (in this case, it is useful to use the Lattice-Boltzman method as a numerical approach [80]). 

In order to match the solutions of Navier-Stokes equation to the soluhon of Darcy s equation at the channel-porous media interface, Darcy s equahon is modified to include a viscous force term in the momentum equation and this is given by the Brinkman s equation (Martys, 2001 Martys et al., 1994) as... 

Notice that Brinkman s equation includes both the pressure force and the viscous force terms. The effective viscosity for the slower-moving fluid in the porous media is selected such that continuity in shear stress is maintained at the interface between the faster-moving gas flow in the channel and the slower-moving gas flow in porous electrode. The continuity in shear stress at the interface is given as... 

For Brinkman s equation, additional boundary condition is given in terms of continuity in shear stress as... 

For the porous electrode-gas diffusion layer with bulk fluid motion given by Darcy or Brinkman s equation, the governing species transport is given as... 

Martys, N., D. P. Bentz and E. J. Garboczi. Computer simulation study of the effective viscosity in Brinkman s equation. Physics of Fluids 6(4) 1434-1438,1994. 

The convection velocity, u, in the porous electrolyte structure is given either by Darcy s law equation (Equation 6.17) or by Brinkman s equation (Equation 6.18) as discussed in Chapter 6. 

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. 

The distribution of flow velocity in the cross-section (see Plate I, line y=0) of tube and yarn bobbin, where the porous medium is described by Brinkman s equation. 

Veiocity profiies in different cross-section lines (see Plate I, y=0.06, 0, and -0.06), where the flow in the porous package is defined by Brinkman s equation. 

The mathematical model presented in Chapter 4 offers the possibility of describing the flow velocity distribution within the package during the dyeing. Figures 6.5 and 6.6 demonstrate the velocity distribution across the cross-section of the tube and yam assembly (see Plate I, line y=0), under different liquor inflow rates, where the flow in porous media is described by Darcy s law and Brinkman s equation, respectively. 

The simulation results demonstrate that Brinkman s equations extend Darcy s law to include a term that accounts for the viscous transport in the momentum balance and introduce the velocities in the spatial directions as dependent variables. This approach seems to be more robust than Darcy s law, since it is valid over a wider range of flow rates, and for different permeabilities of the porous media. 

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