Packings hydraulic radius

Figure 6. Separation factor-particle diameter tehavior as a function of packing diameter for the pore-partitioning model. Parameters are the same as in Figure 3 with the exception of the interstitial capillary radius which was computed from the hed hydraulic radius (Equation 11 (7.) with void fraction = 0.358).

With decreasing packing size in SEC columns, the probability of physical entrapment of macromolecules increases. To estimate the molecular weight limit above which ultrafiltration will occur, we must first calculate an average radius of the interstices formed in a packed bed. This is done by assuming that the packed column consists of a bundle of capillaries in which the capillary radius can be estimated from the bed hydraulic radius ... 

Table 1. Molecular Weight Equivalent to the Hydraulic Radius of a Column Packed with 2-, 5-, and 10-pm Particles...

What is the hydraulic radius for a powder of close-packed spherical particles of radius R 74% of the volume is filled by the particles. 

The general approach is to define the traditional hydraulic radius, but for a more complex cross section for flow. For a porous medium, it is equivalent to the ratio of void volume to surface area, which is known for certain geometries, such as a packing of spheres ... 

Previously it was indicated that for noncircular conduits Fig, 6.10 (the friction factor plot) could be used if we replaced the diameter in both the friction factor and the Reynolds number with 4 times the hydraulic radius (HR). The hydraulic radius is the cross-sectional area perpendicular to flow, divided by the wetted perimeter. For a uniform duct this is a constant. For a packed bed it varies from point to point. But if we multiply both the cross-sectional area and the perimeter by the length of the bed, it becomes... 

If we multiply th by 1, that is, by LIL, then Th becomes the ratio of the void volume of the packing material to the total surface area of the packing material [8]. In other words, the hydraulic radius can now be defined in terms of the packing material [9]... 

Equation (95) has been found to provide a good estimation of permeability for packed beds of voidage between 0.26 and 0.80. Carman (1956) also found that if the hydraulic radius was used to replace the particle radius in Eq. (95), the equation was also good for mixtures of different particle sizes. The hydraulic radius for a bed of spherical particles can be ealeulated from... 

A hydraulic radius model assumes the fluid path in the void spaces in a packed bed to be equivalent to an assembly of the tubes whose radius is taken as the hydraulic radius of the void spaces Then the resultant relation of mass transfer coefficient is given as... 

Basically, the hydraulic radius approach is used to derive appropriate flow equation. For a packed bed we have... 

The constant relating pressure drop to V po in Equation 1-4 for the dry line is actually the summation of the effects of packing shape factor, bed void fraction, and hydraulic radius of the packing as indicated in Equation 1-2. This constant can be determined from dry line pressure drop measurements for any particular type and size of packing, as long as the gas-phase flow is turbulent. 

In this model, the equivalent diameter of the packing is taken to be four times the hydraulic radius where the triangular flow channel is considered to have two metal sides and one open side. Thus, if the included angle between the two sides of the flow channel is 90° in a plane perpendicular to the longitudinal axis of the flow channel, the hydraulic radius is 0.25 times the length of one side of the flow channel [11]. 

The flow in a packed bed, where the packing may be spherical, c indrical, etc., is quite complex (Figure 6.1.1(b)). However, it is often modeled as a collection of cylindrical capillaries of hydraulic radius R/, and length L, which is the packed bed length. Let e be the fractional void volume of the bed, Op be the total particle surface area per particle volume, v be the actual interstitial velocity in the void volume between particles and dp (= 2rp) be the mean particle diameter. Then the superficial velocity Vo based on the empty cross section of the packed bed is defined as... 

Another important value is the equivalent (hydraulic) diametra- of the packing dh. There are two different possibilities to ded this value. Hie first of them comes fiom the consideration of the packing structure in terms of the called iimer problem, i.e. when the packing is considered as a s tem of connected channels with different dimensions. The equivalent diameter by this model is equal to the average hydraulic diameter of these chaimels which is equal to 4 times file hydraulic radius, cm... 

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