Forchheimer equation

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] ... 

For each thickness, at least 10 different flow rate measurements were obtained in order to cover the range of flow rates that a DL experiences during normal fuel cell operation. To obfain fhe corresponding permeabilify, fhe pressure drop resulfs were ploffed as a function of the mass flow rate. After this, the Forchheimer equation was fitted to the plotted data to determine the viscous and inertial permeabilities. As expected, the in-plane permeabilities of each sample DL maferial decreased when the compression pressure was increased. It is also important to mention that these tests were performed in two perpendicular directions for each sample in order to determine whether any anisotropy existed due to fiber orienfation. 

Gurau ef al. [129] presented another apparatus used to measure the in-plane viscous and inertial permeability coefficients. In their method, an annular DL sample was placed between an upper and lower fixture. The gas entered the upper fixture and was then forced fhrough fhe DL info fhe ouflef porfs (open to the atmosphere). A strain sensor was located in the upper fixture in order to determine the thickness of fhe DL (i.e., deformation) because fhe whole assembly was compressed to a determined pressure. In fhis mefhod, the flow rate, temperatures in both fixtures, and pressures were monitored in each test. Once the data were collected, the in-plane permeability was determined from the Forchheimer equation by application of fhe leasf squares fit analysis method. 

In order to determine the viscous and inert through-plane gas permeabilities of diffusion layers at varied compression pressures, Gostick et al. [212] designed a simple method in which a circular specimen was sandwiched between two plates that have orifices in the middle, aligned with the location of the material. Pressurized air entered the upper plate, flowed through the DL, and exited the lower plate. The pressure drop between the inlet and the outlet was recorded for at least ten different flow rates for each sample. The inert and viscous permeabilities were then determined by fitting the Forchheimer equation to the pressure drop versus flow rate data as explained earlier. 

The average pore velocity is related to the flow velocity through the Dupuit-Forchheimer equations 44),... 

Beginning at approximately Re = I, inertial forces become significant, and the linear relationship is no longer valid. The Forchheimer equation addresses this limitation by including a quadratic velocity term ... 

Permeability is the material property that most influences the drying process of refractory castables [43—45]. The permeability of compressible fluids flowing through rigid and homogeneous porous media is described by the Forchheimer equation, which includes a quadratic term for the flow rate q. For small changes in pressure, the Forchheimer s equation leads to Darcy s law ... 

Here we have used the intrinsic average concentration since this is most closely related to the concentration in the fluid phase, and we have used the superficial average velocity since this is the quantity that normally appears in Darcy s law (Whitaker, 1999) or the Forchheimer equation (Whitaker, 1996). Use of equation 1.87 in equation 1.84 leads to... 

Whitaker S. 1996. The Forchheimer equation A theoretical development. Transport Porous Med., 25, 27-61. 

Huang, H. and Ayoub, J. 2006. Applicability of the Forchheimer Equation for Non-Darcy Flow in Porous Media. Paper SPE 102715 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 24-27 September. DPI 10.2118/102715-MS. 

The form of equation 4.16 is similar to that of equation 4.17 proposed by Forchheimer(8) who suggested that the resistance to flow should be considered in two parts that due to the viscous drag at the surface of the particles, and that due to loss in turbulent eddies and at the sudden changes in the cross-section of the channels. Thus ... 

Up to this point, we have only considered cases where inertia is negligible. If inertia is important, then one can generalize Forchheimer s equation [15,16,24,25] to obtain an expression for fd as follows ... 

Xu et al. (2006b) have successfully introduced Brinkman-Forchheimer-extended Darcy equation in order to solve the performance of molten carbonate fuel cell. As a verification of REV method, Poiseuille flow profiles in the porous media modifying LBM with... 

These equations represent the Brinkman extension of the Darcy model. The Forchheimer and Brinkman extensions of the basic Darcy model often must be simultaneously used. 

Equation 12 is valid for both Darcy and Forchheimer flow regimes. Figure 1 shows a general sketch of fv variation with Rem for a porous medium. 

Figure 20 shows the calculated pressure drop factor and the experimental values. We observe that the model of Liu et al. (32) predicts the experimental pressure drop both in the Darcy s flow regime, the transition, and the Forchheimer regimes. The two-dimensional model gives a much better prediction than that using the one-dimensional model. The Ergun equation significantly overpredicts the experimental data. 

The permeability of the foam monolith was calculated using the Darcy-Forchheimer law and the permeability and Forchheimer coefficient of the filter are determined by regression using equation (2). Table 4 summarizes the results for each filter sample tested. 

Reynolds number, is borehole radius, /i is characteristic vertical sizes of the model area K permeability of the media, p and pi fluid viscosity and density, respectively. For the typical conditions of the geothermal reservoir exploitation, /i =100m, PI,= l )00kg/m 9=200-lO mVs, p=3-10S-N/m% 7-w=0.1m and /k=10 m we have Re=10 Hence, flow is laminar and the Forchheimer drag force can be neglected in the model. In this case the linear Darcy momentum equations can be employed, namely, Uf =-(Kj i)(dPIdf), where/=x, y, z. 

Dupuit 1863) and Forchheimer (Forchheimer 1901) to generalize the flow equation, i.e. 

Equation [1.43] for the case of high Reynolds numbers has a correction factor known as Forchheimer coefficient, F ... 

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