Darcy, definition

Equation 3.11 is due to Blasius(6) and the others are derived from considerations of velocity profile. In addition to the Moody friction factor / = 8R/pu2, the Fanning or Darcy friction factor / = 2R/pu2 is often used. It is extremely important therefore to be clear about the exact definition of the friction factor when using this term in calculating head losses due to friction. 

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. 

Although pV2/2 represents kinetic energy per unit volume, pV2 is also the flux of momentum carried by the fluid along the conduit. The latter interpretation is more logical in Eq. (5-50), because rw is also a flux of momentum from the fluid to the tube wall. However, the conventional definition includes the (arbitrary) factor i. Other definitions of the pipe friction factor are in use that are some multiple of the Fanning friction factor. For example, the Darcy friction factor, which is equal to 4/, is used frequently by mechanical and civil engineers. Thus, it is important to know which definition is implied when data for friction factors are used. 

Because we do not have the slope to determine u, we will use the following equation that results from the definition of the Darcy-Weisbach friction factor ... 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

Units of Permeability—Permeability as defined by Darcy s law (Eq 13-4) has the units [(P] = [L2]. Expressed in terms of the definition given above, using cgs units... 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

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

Another measure of length is the equivalent hydraulic radius, rh, determined inversely from saturated or unsaturated flow experiments. By comparing Darcy s equation with Poiseuille s law, and invoking a capillary bundle model, one obtains the following definition of rh (Kutflek Nielsen, 1994),... 

In this equation, known as the Darcy Equation, and which is applied in hydrogeology for calculating advective fluxes in groundwater, vjm s ] denotes the velocity with which a particle/solute crosses a definite distance in aqueous sediments. Ap refers to the pressure altitude measured in meters of water column (10 Pa = 1 bar 750 mm Hg 10.2 m water column) and Ax [m] is the distance across which the pressure difference is measured. In the example shown above ((]) = 0.77, k = 7-10 m s ), this distance amounts to 4 m. Insertion into Equation 3.30 yields ... 

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. 

Thus, if the mass conservation requirement div q = 0 holds, and it does for incompressible flows where q is the Darcy velocity, the identity guarantees that we can represent q = curl V, which reduces to our planar in that limit. The use of a three-dimensional T, however, does not render path tracing any easier than dealing with three velocity functions (in two-dimensional problems, a single streamfunction suffices). Thus, we will not pursue any further discussion. But the idea of streamlines or pathlines as tangents locally parallel to the velocity vector is still attractive, and kinematically, we would expect a definition along the lines of dx/dt u, dy/dt v, and dz/dt w, where denotes proportionality. Let us consider an interface located anywhere within a flow, that is, any surface marked by red dye, and describe it by the locus of points f(x,y,z,t) = 0 (4-103)... 

Another definition of friction factor is that used in the Darcy equation for pressure drop. The Darcy friction factor, f, is defined as... 

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