Voidage, total

The total voidage i in a packed bed (outside and inside particles) is... 

In order to evaluate the total fluid flowrate, the Integra) on tlie RHS of equation 2.73 must be evaiuated over pressure range using a known (usuaiiy measured) function of voidage with pressure. 

Only a few investigations concerned with the measurement of gas holdup and residence-time distribution have been reported. The information regarding liquid holdup, which will be discussed in the following section, is considerably more abundant in some cases, values of gas holdup can be deduced from the reported data on liquid holdup and total voidage. 

Piret et al. measured liquid holdup in a column of 2J-ft diameter and 6-ft packed height, packed with graded round gravel of lj-in. size, the total voidage of the bed being 38.8%. The fluid media, air and water, were in countercurrent flow. The liquid holdup was found to increase markedly with liquid flow rate, but was independent of gas flow rate below the loading point. Above the loading point, an increase of liquid hold-up with gas flow rate was observed. 

The accumulated liquid fills part of the originally free gas volume lowering the porosity of the filter to a certain extent, but typically not totally due to the transport of the liquid, if the liquid is not too viscous. On the contrary, there is no such limitation for accumulated solid particles which can fill the voidage until the filter is clogged. 

The bubble voidage in the draft tube, ebr, was calculated on the basis of the velocity of a rising gas slug relative to its surrounding solids. The total gas superficial velocity in the draft tube, lJjn can be derived to be... 

The bed voidage, s, is the ratio of the interpartide void space to the total volume V ... 

A binary suspension consists of equal masses of spherical particles of the same shape and density whose free falling velocities in the liquid are 1 mm/s and 2 mm/s, respectively. The system is initially well mixed and the total volumetric concentration of solids is 0.2. As sedimentation proceeds, a sharp interface forms between the clear liquid and suspension consisting only of small particles, and a second interface separates the suspension of fines from the mixed suspension. Using a suitable model for the behaviour of the system, estimate the falling rates of the two interfaces. It may be assumed that the sedimentation velocity uc in a concentrated suspension of voidage e is related to the free falling velocity u0 of the particles by ... 

A slurry containing 100 kg of whiting/m3 of water, is filtered in a plate and frame press, which takes 900 s to dismantle, clean and re-assemble. If the filter cake is incompressible and has a voidage of 0.4, what is the optimum thickness of cake for a filtration pressure of 1000 kN/m2 The density of the whiting is 3000 kg/m3. If the cake is washed at 500 kN/m2 and the total volume of wash water employed is 25 per cent of that of the filtrate, how is the optimum thickness of cake affected The resistance of the filter medium may be neglected and the viscosity of water is 1 mN s/m2. In an experiment, a pressure of 165 kN/m2 produced a flow of water of 0.02 cm3/s though a centimetre cube of filter cake. 

In a packed bed of unit volume, the volumes occupied by the voids and the solid particles are e and (1 — e) respectively where e is the voidage fraction or porosity of the bed. Let S0 be the surface area per unit volume of the solid material in the bed. Thus the total surface area in a packed bed of unit volume is (1 — e)S0. 

For relatively coarse particles, the rates of fall of the interface between (a) and (b) and between (b) and (c) can be calculated approximately if the relation between sedimentation velocity and voidage, or concentration, is given by equation 5.76. Richardson and Shabi 65) have shown that, in a suspension of particles of mixed sizes, it is the total concentration which controls the sedimentation rate of each species. 

Because there is no net displacement of liquid at a voidage of 0.854, corresponding to a total volumetric concentration of particles of 14.6 per cent, the concentration of each zone of suspension, after separation, should be equal. This has been confirmed within the limits of experimental accuracy. It has also been confirmed experimentally that upward movement of the polystyrene particles does not take place at concentrations less than 8 per cent (e > 0.92). 

In fixed-beds, the space-time as defined above is the superficial space time, as it is based on the total volume of the bed. The real space-time for a bed of bed voidage e is... 

In the analysis above, the void volume of the fixed bed (sVR) is considered to be fully filled with the fluid phase, i.e. the fluid holdup based on the total volume of the bed he t is equal to the bed voidage e. While this is expected in the case of a gas as fluid, it is not always true in the case of a liquid, especially in the case of downflow operation. In the case of incomplete filling of the bed with the fluid phase, the active bed volume is lower, and thus a portion of the solid phase is not in contact with the fluid, or in other words, is inactive in terms of reaction. 

Liquid holdup based on the total volume of the bed (empty bed volume) occupied by the liquid he and liquid holdup based on the void volume of the bed (bed voidage) occupied by the liquid are related as follows ... 

In a reactor completely filled with liquid, the wetting efficiency is 100% or, in other words, the external wetting of the catalyst is complete (Burghardt et al., 1995). While it is true that when a fixed bed is completely filled with liquid wetting is complete (wetting efficiency is unity), the opposite is not true in a trickle bed, a portion of the bed voids will be always occupied by the gas phase. Thus, while in a well-operated trickle bed the wetting efficiency could be unity, its total liquid holdup based on the void volume is always lower than the bed voidage, i.e. the bed is never completely filled with liquid. 

The mass transfer factor has also been correlated as a function of the Reynolds number only and thus taking account only of hydrodynamic conditions. If e is the voidage of the packed bed and the total volume occupied by all of the catalyst pellets is Vp, then the total reactor volume is Vp/(l - e). Hence the rate of mass transfer of component A per unit volume of reactor is NASx(l - e)/Vp. If we now consider a case in which only external mass transfer controls the overall reaction rate we have ... 

Figure 20 Predicted flow regime diagram of the industrial MIP reactor, with solids flux as a function of the imposed total pressure drop at fixed gas flow rate. The snapshots of voidage profile refer to the transition, from left to right, the dilute transport, choking transition in between with different solids inventory, to the dense fluidization (Lu et al., 2007).

Denote Aa as the pore cross-sectional area and A as the total cross-sectional area of the column. The voidage is... 

Knowing the voidage, the number of particles can be determined. The total volume occupied by the particles in the bed is equal to ... 

The bed void volume available for flow and for gas and liquid holdup is determined by the particle size distribution and shape, the particle porosity, and the packing effectiveness. The total voidage and the total liquid holdup can be divided into external and internal terms corresponding to interparticle (bed) and intraparticle (porosity) voidage. The external liquid holdup is further subdivided into static holdup eLs (holdup remaining after bed draining due to surface tension forces) and dynamic holdup eLrf. Additional expressions for the liquid holdup are the pore fillup Ft and the liquid saturation SL ... 

The operation of this pilot plant combustor is similar to that of the Type B fast fluidized bed in Chapter 3, Section II. Figure 16 (Li et al, 1991) shows the axial profiles of bed voidage for different gas velocities and total pressure drops across the fast fluidized combustor, demonstrating the... 

The determination of the gas or liquid holdup is largely made from RTD measurements. Some investigators have measured the RTD of the gas phase, while others have studied the liquid phase. It should be noted that, in principle, the holdup of only one phase is required because the holdup for the other phase can be calculated if the total voidage is known. 

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