Erguns relation

Darcy s law is vahd only for a sufficiently slow flow, which is laminar inside the porous medium. The validity criterion for Darcy s law is expressed by a Reynolds number, written using the length scale -Jk that characterizes the porous medium. The following criterionfor the validity of Darcy s law is generally agreed upon  

For a Reynolds numbers exceeding the value of 10, Ergun s relation  

Ergun s relation is applied not only to very porous beds but also to non-deposited granular media such as fluidized beds, as we study in Chapter 15. 

Draining by pressing is a common operation for a person who presses a sponge in order to wring the maximum possible amount of liquid out of it. It is also used in industrial processes aiming to reduce the water content in a sludge. From a mechanical point of view, modeling the filtration of the liqnid within the porous medium (Darcy s law) needs be complemented, in order to take into account the 

Formalizing the problem reqnires several steps, different in nature, which ate treated in seqnence. 

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The measurement technique already contains the possibility of calculating the minimum fluidization velocity Mmf. The pressure drop in flow through the polydisperse fixed bed at the point u = mf, given for example by the Ergun relation [20], is set equal to the... 

For the matrices obeying the modified Ergun relation we have... 

It may be important to know the pressure drop across a packed bed in relation to the gas flow. The Ergun relation describes this effect for near-spherical particles, for a range of void fractions (0.3 < e < 0.5) (Ergun, 1952) ... 

AP equation arising from simultaneous turbulent kinetie and viseous energy losses that is applieable to all flow types. Ergun s equation relates the pressure drop per unit depth of paeked bed to eharaeteristies sueh as veloeity, fluid density, viseosity, size, shape, surfaee of the granular solids, and void fraetion. The original Ergun equation is ... 

The next step is to eharaeterize the resistanees offered by the porous eatalyst bed and support sereens. Several eorrelations relating the pressure drop through porous beds and veloeity and bed eharaeteristies are available. We seleet an Ergun equation to represent the resistanee of eatalyst bed ... 

Expression in Eq. (19) is within 8% of all simulation data up to Re — 1000. Since this relation has been derived very recently (Beetstra et al., 2006), it has not been applied yet in the higher scale models discussed in Sections III and IV. However, the expression by Hill et al. in Eq. (47) derived from similar type of LBM simulations is consistent with our data, in particular when compared to the large deviations with the Ergun and Wen and Yu equations. So, we expect that the simulation results presented in Section IV.F using the Hill et al. correlation will not be very different from the results that would be obtained with expression in Eq. (19). A more detailed account of the derivation of expression in Eq. (19) and a comparison with other drag-force relations can be found in Ref. Beetstra et al. (2006). 

Note that the validity of both the Ergun and Wen and Yu equations has recently been questioned on the basis of LB data, and alternative drag-force correlations have been proposed. From LB simulations, Hill et al. (2001a, b) suggest the following relation for Stokes flow (lim Rea->0) ... 

This section is a continuation of Section 21.3.2 dealing with pressure drop (-AP) for flow through a fixed bed of solid particles. Here, we make further use of the Ergun equation for estimating the minimum superficial fluidization velocity, ump In addition, by analogous treatment for free fall of a single particle, we develop a means for estimating terminal velocity, ur as a quantity related to elutriation and entrainment. 

The minimum fluidising velocity, umf, may be expressed in terms of the free-falling velocity o of the particles in the fluid. The Ergun equation (equation 6.11) relates the Galileo number Ga to the Reynolds number Re mj in terms of the voidage < , / at the incipient fluidisation point. 

The Ergun equation relates the pressure drop in a packed bed to the flow rate and the properties of particle and gas. However, the application of this equation has been extended beyond the limits of fixed bed systems since it was first formulated in 1951. Thus, a detailed account of the origin of this equation is necessary. 

The generalized relation for the pressure drop for flows through a packed bed was formulated by Ergun (1952). The pressure loss was considered to be caused by simultaneous kinetic and viscous energy losses. In Ergun s formulation, four factors contribute to the pressure drop. They are (1) fluid flow rate, (2) properties of the fluid (such as viscosity and density), (3) closeness (such as porosity) and orientation of packing, and (4) size, shape, and surface of the solid particles. 

These workers, who originally suggested the above approach, made use of the Ergun (E3) packed-bed equation for relating Umt to gas and solids properties by putting APfH in Ergun s equation equal to (ps — Pi) (1 —... 

The next and most important step is to characterize the resistance offered by the porous catalyst bed and support screens. Several correlations relating the pressure drop through porous beds and velocity and bed characteristics are available (Carman, 1937 Ergun, 1952 Mehta and Hawley, 1969). The Ergun equation is one that is widely used to represent the resistance of a catalyst bed, and has the form ... 

If a liquid is passed vertically upward through a bed of uniform particles, the pressure drop, AP, increases with an increase in the superficial liquid velocity, The relation between pressure drop and velocity is the same as for a fixed bed, as indicated by the following equation (Ergun, 1952) ... 

Consequently, the reactor model is constituted by a system of N+1 equations, where N is the number of chemical species present in the system (NO, NO2, N2 and O2, neglecting the presence of N2O N = 4) and another unknown variable is pressure. The equations are one momentum balance (in the form of simplified Ergun Law), and four mass balance relationships. The presence of NO2 among the reaction products has been related to the catalytic activity of Cu-ZSM5 towards the oxidation of NO to NO2, as revealed by our previous investigation in similar experimental conditions [7], as well as by the present results (Fig. 1). It has been hypothesised that reaction (2) proceeds in parallel to NO decomposition, having not assumed that NO2 formation is responsible for copper reduction from Cu (inactive in decomposing NO) to Cu (the active site), as also proposed by some author [20-21,23]. 

The superficial fluid velocity in the bed follows Ergun s relation locally. 

The interphase momentum transfer coefficient P is frequently modeled by a combination of the Ergun equation and the Wen and Yu correlation, but in this model, the improved drag relation by Beetstra et al. (2007), based... 

Flow through resistive porous elements has been studied by many in the particle filtration community to determine basic relations and empirical correlations (Ergun, 1952 Jones and Krier, 1982 Laws and Livesey, 1978 Munson, 1988 Brundrett, 1993 Olbricht, 1996 Sodre and Parise, 1997 Wakeland and Keolian, 2003 Wu et al 2005 Valli et al 2009). A detailed and rigorous review of previous analytical and numerical solutions in porous pipe, annulus, and channel flow is reserved in Appendix F only highlights are presented here. Porous channel flow is classified by the size of flow within the channel (laminar or turbulent), the number of porous walls (one or two), the size (small, large, arbitrary), and nature (uniform or variable) of injection into the porous element, the type of transverse and axial boundary conditions at the porous surface (suction or injection), and whether or not there is heat transfer and/or electrical or magnetic component, where the injection Reynolds number is defined as ... 

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