Darcy coefficient

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

Finally, the general expression of the Darcy coefficient including singularities is as follows ... 

Keywords aqueous solution, ions, coupling, Darcy coefficient, free energy, stress, strain,... 

Figure 5. Variation of Darcy coefficient K, with water content.

Taking into account the value of V, (Equation (23)) and the definition of tlie Knudsen number, the Darcy coefficient becomes ... 

An error of 2% on a channel dimension can lead to a 14 % error in the Darcy coefficient determination. It is essential to use an adapted instrumentation to measure the geometrical characteristics of a channel. Sometimes, the cross section may not be the same from one end of a channel to the other and, if necessary, the manufacturer s data must be verified carefully. An uncertainty analysis on the Poiseuille number determination is given by Celata [8] following the work by Holman [19]... 

Darcy coefficient 4/ (head loss/velodly 2ghfD/(v t)... 

For an uncompressed Toray TGP-H-060 carbon fiber paper, the Darcy coefficient of 5-10 X 10 m has been reported [31]. Approximately the same value has been reported for in-plane flow through the same material compressed to 75% of its original thickness [31]. 

Friction Coefficient. In the design of a heat exchanger, the pumping requirement is an important consideration. For a fully developed laminar flow, the pressure drop inside a tube is inversely proportional to the fourth power of the inside tube diameter. For a turbulent flow, the pressure drop is inversely proportional to D where n Hes between 4.8 and 5. In general, the internal tube diameter, plays the most important role in the deterrnination of the pumping requirement. It can be calculated using the Darcy friction coefficient,, defined as... 

Table 5. Correlations for Heat-Transfer and Darcy Friction Coefficients for Noncircular Laminar Duct Flow ...

This formula is another variation on the Affinity Laws. Monsieur s Darcy and VVeisbach were hydraulic civil engineers in France in the mid 1850s (some 50 years before Mr. H VV). They based their formulas on friction losses of water moving in open canals. They applied other friction coefficients from some private experimentation, and developed their formulas for friction losses in closed aqueduct tubes. Through the years, their coefficients have evolved to incorporate the concepts of laminar and turbulent flow, variations in viscosity, temperature, and even piping with non uniform (rough) internal. surface finishes. With. so many variables and coefficients, the D/W formula only became practical and popular after the invention of the electronic calculator. The D/W forntula is extensive and eomplicated, compared to the empirieal estimations of Mr. H W. 

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 ... 

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. 

Sometimes, however, A is expressed as [jtd2]/[4( 144)] with the ji/4(144) buried into an overall coefficient. For example, Crane14 has a solved problem that uses the Darcy equation form ... 

Figure 26.9 illustrates Darcy s law, the basic equation used to describe the flow of fluids through porous materials. In Darcy s law, the coefficient k, hydraulic conductivity, is often called the coefficient of permeability by civil engineers. 

Transmissivity is simply the coefficient of permeability, or the hydraulic conductivity (k), within the plane of the material multiplied by the thickness (T) of the material. Because the compressibility of some polymeric materials is very high, the thickness of the material needs to be taken into account. Darcy s law, expressed by the equation Q = kiA, is used to calculate the rate of flow, with transmissivity equal to kT and i equal to the hydraulic gradient (see Figure 26.22) ... 

It is often easy to measure the flux density, e.g., using a flowmeter, and then determine the hydraulic conductivity or diffusion coefficient by dividing the flux by the driving force. One of the most difficult problems is determining how to represent the driving force. The symbol V is called an operator, which signifies that some mathematical operation is to be performed upon whatever function follows. V means to take the gradient with respect to distance. For Darcy s law under saturated... 

In other cases, researchers assume that the inertial resistance to flow in DLs adjacent to conventional flow fields is negligible and they tend to lump both viscous and inertial coefficients together. This may not be a correct assumption, especially when dealing with flow fields like the interdigitated design [129,212], in which higher velocities are experienced in the pores of the DL. The Forchheimer equation is an extension of Darcy s law and takes into account the inertial resistance at high velocities [211,213] ... 

With these data and Darcy s law, the in-plane viscous permeabilihes were determined. Only the viscous permeability coefficient was determined because it was claimed that the inertial component was undetectable within the error limits of measuremenf for fhese fesfs. If is imporfanf to mention fhaf fhis technique could also be used to measure fhe permeabilify of diffusion layers wifh different fluids, such as liquid wafer. 

Chang et al. [183] presented a similar design in which two discs (with orifices in the middle) were used to compress the sample material. Pressurized air (without any moisture) was then passed through the orifices of fhe discs toward the sample DL, which then flowed peripherally to the atmosphere. The two discs were compressed in order to see how the permeability of the DL changed as a function of the clamping pressure. The permeability coefficient was solved using Darcy s law thus, only the viscous in-plane permeability was taken into account. Other, similar techniques can be found in the literature [215-217]. 

Other methods to study the through-plane permeabilities were presented by Chang et al. [183] and Williams et al. [90]. However, these methods only determined the viscous permeability coefficient with Darcy s law and did not take into account the inertial component of the permeability. 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

The physics of thermal conduction and storage are, in fact, directly analogous to those of groundwater flow. Thermal conductivity (kT) and hydraulic conductivity (k) are analogous, as are heat capacity and storage coefficient and temperature (7) and hydraulic head (h). Indeed, heat flow (H) is estimated by an analogous equation to Darcy s Law ... 

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