Number Darcy-modified

Using these values the Darcy-modified Rayleigh number based on the height of the bottle... 

The solution discussed above has, as parameters, Raw, the Darcy-modified Rayleigh number based on the enclosure width, and A. the enclosure aspect ratio HfW. Some typical results computed with the above program are shown in Figs. 10.29 and 10.30. 

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. 

The Darcy-modified Rayleigh number, based on characteristic dimension L, is defined as follows... 

In this equation, ( ) is the porosity in fraction, and u is the Darcy velocity of the displacing fluid. The velocity used by Abrams (1975) is v/[( )(Soi - Sm)]. He also modified the capillary number by multiplying the viscosity ratio (liw/lio)°-" ... 

Analyze the linear stability of the fluid layer. The analysis is similar to the classical Rayleigh-Benard problem, but simpler because the Navier-Stokes equations are replaced with Darcy s law. It can be solved analytically. In place of the Rayleigh number, you should find that the stability depends on a modified Rayleigh number,... 

Figures 14 and 15 show the normalized pressure drop factor for a densely packed bed of monosized spherical particles. For Rem < 7,fv is fairly independent of Rern, and at high Rem values, it increases fairly linearly with Rem. The data points are the experimental results taken from Fand et al. (110), where the bed diameter is D = 86.6 mm and the particle diameter is ds = 3.072 mm. One can observe that the 2-dimen-sional model of Liu et al. (32), referred to as equation 107, agrees with the experimental data fairly well in the whole range of the modified Reynolds number. From Figure 14, one observes a smooth transition from the Darcy s flow to Forchheirner flow regime. The one-dimensional model of Liu et al. (32) (i.e., equation 106) showed only slightly smaller fv value. Hence, the no-slip effect or two-dimensional effect for this bed is small. As shown in Figures 14 and 15, the Ergun equation consistently underpredicts the pressure drop. The deviation becomes larger when flow rate is increased.

In macrolevel analysis, capillary number can be modified while retaining Darcy s law to describe the bulk impregnation of the resin into the fiber preforms. It is useful to point out that the anisotropic nature of the fiber preforms will now appear in the permeability tensor and we can expect some directionality of the capillary pressure influence, which is not necessary in almost all other research areas, where capillary pressure is used. Thus... 

Flow in the laminar regime is often characterized by a friction loss factor, which is 64/Re for the Darcy factor or 16/Re for the Fanning factor (this topic wiU be discussed in more details in Chapter 2). As a result, the losses in the laminar regime appear to be a linear function of speed, whereas in the turbulent regime they are proportional to the square of the speed. As we wiU see in Chapter 5, researchers have struggled with special definitions of a modified Reynolds number for non-Newtonian flows. 

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. 

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