Darcy liquid flow

The driving force behind the liquid transport has been determined by Kramer and Bubeck to be capillary pressure [17]. The velocity of liquid flow V is expressed by Darcy s equation... 

A pressure drop, VP, along the pores leads to liquid flow. The mass flux in the pores of the green body due to liquid flow is given by Darcy s law ... 

Microfiltration and ultrafiltration are the two main filtration techniques for which ceramic membranes have been widely used to date. As described in Section 6.2.1.2, MF and UF ceramic membranes exhibit macro- and mesoporous structure, respectively, which result from packing and sintering of ceramic particles. Liquid flow in such porous media is convective in nature and the simplest description of permeation flux, J, is given by the Darcy s equation [20] ... 

Combining (9.21) with the Kelvin equation (9.18) and using Darcy s law for liquid flow through porous media (as similarly done before for gas flow, see Eq. (9.1)) yields an expression for the gas flux Jd3 of capillary condensate in case 3 of Fig. 9.9 ... 

It is worth mentioning that while Darcy s law is often used to describe gas and liquid flow under heterogenous conditions, its validity under extended conditions is not accepted without debate (e.g., Refs. ). Consideration should therefore be given to the particular conditions under investigation to ensure the applicability of Darcy s law to describe fluid flux. 

Relationship of superficial liquid flow rate q to liquid pressure gradient, Darcy s law. Empirical constitutive equations relating permeability K, specific resisfance a, and soli-dosify (vol fraction of solids) e, to the effective pressure. 

During this shrinkage, the permeability of the gel, which is low, decreases further. While shrinkage continues, the quantity of liquid removed by evaporation is balanced by the liquid flow to the surface. Darcy s law gives the liquid flux J due to the pressure gradient VP ... 

The liquid flow rate in the porous media is given by Darcy s law ... 

The constant drying-rate period lasts as long as the surface is supplied with liquid. Its duration depends strongly on the drying conditions (magnitude of the external flux) and on the medium properties. The liquid flow inside the medium is expressed by Darcy s law (permeability x gradient of liquid pressure). 

A. Noncompressible Fluid Flow. The Darcy formula, Eq. (13), is applicable for noncompressible liquid flow. It applies in all situations with the possible exception of a liquid pressure drop so high that the outlet pressure drops to the vapor pressure of the liquid. This condition has to be avoided by changing the pipe size, decreasing the flow rate, or forcing a back pressure. Re in Eq. (13) is calculated from Eq. (15), the flow velocity from the area of the pipe. A, and the volumetric flow rate, Q, as ... 

The volume shrinkage of the gel during drying induces an increase of its stiffness. At a given time, the solid network is no more compliant and the meniscus recedes in the pores. At this moment, the stress is maximum since the curvature radius corresponds to that of the shrunk pore (assumed cylindrical). Associated to evaporation, the liquid flows from the core of e gel to the surface. This flow is hindered by the solid arms of the gel. A gel is badly permeable because the size of the pores lies mainly within the range 0.2-10 nm indicating that a gel is a mesoporous material. According to the Darcy s law, the liquid flow, /, is related to permeability, D, by the relation ... 

Now, as in the first example of the previous section, we assume a onedimensional, constant density, single liquid flow. For such flows, the constant Darcy velocity is (k/p)(Ap/L), where Ap > 0 is the usual delta p or pressure drop through the core of length L. The corresponding velocity for the present problem is v = (k/p)(Ap/> ) where k is the cake permeability, and pis the filtrate viscosity. Substitution in Equation 17-8a leads to... 

The Darcy-Weisbach Equation is a generally accepted method for calculating friction losses from liquids flowing in full pipes. It recognizes the dependence on pipe diameter, pipe wall roughness, liquid viscosity, and flow velocity. Darcy-Weisbach is a general equation that applies equally well at any flow rate and any incompressible fluid. 

The point is that acidizing in sandstones realistically addresses skin only, which can have a drastic effect on well productivity. As shown in equation (3.1), well production rate, q, can be defined by Darcy s law for steady-state liquid flow in a radial reservoir as a function of permeability, k, and skin, s, among other factors. Again, skin, s, is a multicomponent term representing total skin s), composed of different skin contributors, as given in equation (3.3). 

Though Washburn model [63] is most commonly applied to describe liquid infiltration into textiles, it has got some limitations such as the model cannot be used to model source driven flows without modification and it cannot be applied to describe spreading in two- or three-dimensions. The other equation which is commonly used to model liquid transport in porous material is Richard s equation [64]. Richard s equation is a semi-empirical formulation based on Darcy s law, where the rate of volmnetric liquid flow is proportional to the driving pressure. An essential character of the models based on Richard s equation is that they contain the functional relation between permeability, pressure and moisture. 

The analysis given here is concerned with the movement of fluids through porous media the equation describing motion (momentum balance) is thus of central importance. Following the original work of Hemy Darcy, mathematical descriptions of liquid flow in porous media are based on Darcy s law. This law states that the volumetric flow rate Q of hquid through a specimen of porous material is proportional to the hydrostatic pressure difference Ap across the specimen, inversely proportional to the length of the specimen, and proportional to the cross-sectional area. Darcy s law is expressed simply as ... 

The liquid flow inside the porous layer obeys the Darcy equation... 

Because flashing steam-condensate lines represent two-phase flow, with the quantity of liquid phase depending on die system conditions, these can be designed following the previously described two-phase flow methods. An alternate by Ruskin [28] uses the concept but assumes a single homogeneous phase of fine liquid droplets dispersed in the flashed vapor. Pressure drop was calculated by the Darcy equation ... 

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