Darcy number

Effectively, Eqs. (86) and (87) describe two interpenetrating continua which are thermally coupled. The value of the heat transfer coefficient a depends on the specific shape of the channels considered suitable correlations have been determined for circular or for rectangular channels [100]. In general, the temperature fields obtained from Eqs. (86) and (87) for the solid and the fluid phases are different, in contrast to the assumptions made in most other models for heat transfer in porous media [117]. Kim et al. [118] have used a model similar to that described here to compute the temperature distribution in a micro channel heat sink. They considered various values of the channel width (expressed in dimensionless form as the Darcy number) and various ratios of the solid and fluid thermal conductivity and determined the regimes where major deviations of the fluid temperature from the solid temperature are found. 

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. 

Permeability may vary, for example, for clean gravel, from 10 to 10 m. The Darcy number for permeability is widely used in applications of interest to geophysicists and corresponds to 0.987 x 10 m. ... 

For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness /D also affects the friction factor. Figure 6-9 plots/as a function of Re and /D. Values of for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction fac tor used by Moody Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor ... 

Darcy s law is considered valid for creeping flow where the Reynolds number is less than one. The Reynolds number in open conduit flow is the ratio of inertial to viscous forces and is defined in terms of a characteristic length perpendicular to flow for the system. Using four times the hydraulic radius to replace the length perpendicular to flow and conecting the velocity with porosity yields a Reynolds number in the form ... 

Heterogeneity, nonuniformity and anisotropy are defined as follows. On a macroscopic basis, they imply averaging over elemental volumes of radius e about a point in the media, where e is sufficiently large that Darcy s law can be applied for appropriate Reynolds numbers. In other words, volumes are large relative to that of a single pore. Further, e is the minimum radius that satisfies such a condition. If e is too large, certain nonidealities may be obscured by burying their effects far within the elemental volume. 

Column length and diameter these are set according to the minimum number of plates required and the maximum acceptable pressure-drop, and calculated by taking into account the constraints given by the Van Deemter and Darcy s laws. 

The basis for single-phase and some two-phase friction loss (pressure drop) for fluid flow follows the Darcy and Fanning concepts. The exact transition from laminar or dscous flow to the turbulent condition is variously identified as between a Reynolds number of 2000 and 4000. 

Figure 12.6 Darcy coefficient of pressure drop versus the Reynolds number in a Corning reactor.

Group N6 (or some multiple thereof) is also known as a friction factor (/), because the driving force (AP) is required to overcome friction (i.e., the energy dissipated) in the pipeline (assuming it to be horizontal), and N3 is known as the Reynolds number (N e). There are various definitions of the pipe friction factor, each of which is some multiple of N6 e.g., the Fanning friction factor is N6/2, and the Darcy friction factor is 2N6. The group N4 is also known as the Euler number. 

In practice we measure changes in pressure and salt concentration, rather than chemical potential, and it can be helpful to think of transport coefficients in terms of Darcy flow and diffusion. We define xs = ns/(nw + ns), where nw, ns are the number of moles of water and salt, and write the change in chemical potential of salt across the membrane as... 

At higher Reynolds numbers, the friction factor is affected by the roughness of the surface, measured as the ratio e/D of projections on the surface to the diameter of the pipe. Glass and plastic pipe essentially have = 0. Table 7-11 gives the pipe roughness of various materials. Figure 7-22 shows plots of Darcy friction factor versus Reynolds number for various pipe sizes. Alternatively, an explicit equation for the friction factor is given by [30] ... 

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. 

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