Tube-particle diameter ratio, effect

An experimental evaluation of homogeneous continum models of steady state heat transfer in packed beds of low tube/particle diameter ratio has been carried out. It was found that both axial and radial conduction effects were important in such beds for N j 500, which covers the flow range in many industrial reactors. Heat transfer resistance at the wall was significant, but of secondary importance. 

Deviations from an ideal plug-flow pattern are caused by either wall flow or axial gradients that develop in the direction of flow. The bed void fraction at the reactor wall is likely to be somewhat higher than the void fraction in the catalyst bed, Gb-In order to eliminate the wall effect on the flow pattern, the tube diameter to particle diameter ratio is chosen to be greater than 10 as a general rule, but since the microreactors used for kinetic studies have very small diameters (4-10 mm), and for reasons discussed in the following, a higher ratio of 15 is indicated to minimize the wall effect in laboratory PBRs ... 

Axial gradients may arise as a result of reactant conversion along the catalyst bed, which may be important in integral reactors. The rule of thumb for minimizing axial dispersion effects is concerned with the reactor tube length to particle diameter ratio, which is reported as being at least 50 for first-order reactions and particle Reynolds numbers greater than 10 ... 

The tube-to-particle diameter ratio is usually larger than 5 to assure an appropriate packing at industrial conditions. Values around 3.3 have been used in the pilot plant in Figure 3.6, but such small values require very careful loading. The length parameter in the Re number in Equation (3.23) depends on particle shapes and has also been measured for rings, where a length effect has been found [42]. 

Fixed beds of very low tube-to-particle diameter ratio have also been proposed and studied. For these reactors, the effect of wall and particle shape on bulk voidage becomes important. It will be convenient to... 

Radial profiles of the dimensionless axial velocity. Effect of tube-to-particle diameter ratio at Z =. z Z, = 10 for Rep = 175. From Papageorgiou and Froment [1995]. 

There is no radial velocity, and the axial velocity across the radius of the packed bed is uniform. Schwartz and Smith (1953) found that the velocity across the diameter of a packed bed is not uniform for radial aspect ratios (tube-to-particle diameter) less than about 30, due to the significant effect of the increased void space near the wall where the particles are locally ordered. This result has been verified by Hoiberg et al. (1971) for a packed bed reactor with radial aspect ratio about 50. They considered a radial velocity variation suggested by experimental observations with a sharp peak about 15% greater than the mean fluid velocity situated close to the wall. Simulations using their model showed results virtually identical to those obtained with a uniform velocity profile.3... 

Agnew and Potter [6] did the same as Barkelew for heterogeneous catalytic reactors and presented design diagrams to prevent runaway, including also the parameter of the ratio of the tube to the catalyst particle diameters dt/dp. Burghardt and Warmuzinski [7] considered multiple reactions and also took the heat effect of the secondary reaction into account however, they did not study the selectivities achieved in the reactor. 

With the measurements subject to fluctuations of 20 or 30%, no accurate description of the profile is possible. All that can be said is that with moderate ratios of tube to particle diameter, the maximum velocity is about twice the minimum, and that when the particles are relatively small, the profile is relatively flat near the axis. It is fairly well established that the ratio of the velocity at a given radial position to the average velocity is independent of the average velocity over a wide range. Another observation that is not so easy to understand is that the velocity reaches a maximum one or two particle diameters from the wall. Since the wall does not contribute any more than the packing to the surface per unit volume in the region within one-half particle diameter from the wall, there is no obvious reason for the velocity to drop off farther than some small fraction of a particle diameter from the wall. In any case, all the variations that affect heat transfer close to the wall can be lumped together and accounted for by an effective heat-transfer coefficient. Material transport close to the wall is not very important, because the diffusion barrier at the wall makes the radial variation of concentration small. 

Figure 9 shows the value of k as a function of the ratio of bed diameter over particle diameter, as determined from packed gas chromatographic columns (d). It can be seen that k tends to decrease as the diameter ratio increases, which implies that flow becomes more uniform. Whereas at low ratios k is inevitably high (of the same order of magnitude as in laminar flow through empty tubes) due to the wall effect, at higher diameter ratios k can vary more widely since its value depends upon whether the colunm is well or badly packed. 

Figure 6.1 Effect of velocity ratio on concentration ratio for a sampling tube oriented in the direction of the mainstream flow. The curves are approximate representations of the data of various experimenters for unit density particles of diameters (in /rn i) as indicated. The displacement of the point tig/iita = 1 from = I results from the finite particle diameter. The curves apply to a...

Table IV gives a summary of the packed beds that we made use of in this section. The term Cm< reflects the two-dimensional effects for Rem = 0. A value of zero, for 100(Cm / — 1), would indicate no two-dimensional effects. We can observe that the wall effects on the viscous term, Cw2, range from about 6% for the experimental data of Fand et al. (110) to 274% for the packed bed of Liu et al. (32) with a ds/D = 0.7123 as used here. The wall effects on the inertial term, Cwh range from around 1 to 19%. The two-dimensional effects are also significant for the packed beds of large particle to tube diameter ratios. The 2-dimensional model of Liu et al. (32) predicts quite well over a wide range of wall effects. In contrast, the wall modified Ergun equation significantly underpredicts at low porosity (Figures 15 and 19) and overpredicts at high porosity (Figures 16 and 17) the experimental data.

In this chapter it will be shown how 2D models can be used to predict the extent of external mass transfer limitations and their effect on the reactor performance. Also the effect of a radial porosity profile (important where the ratio of the tube diameter over the particle diameter is smaller than about 10) can be included in the model. 

The length and diameter of the tube and the particle size (hydratdic diameter) also affect flow distrihution within the packed tube. If the ratio of the tube diameter to that of the particle diameter is above 30, radial variations in velocity can be n lected, and plug (piston) flow behavior can be assumed. The ratio of the tube length to particle diameter is also important if this ratio exceeds 50, axial dispersion and axial heat conduction effects can be ignored. These efiects bring notable simplifications into the modeling of PBRs, which are discussed in Chapter 3. 

Radial variations in bed structure for a shallow reactor with a low ratio of the tube to particle diameter, undesirable wall effects (bypassing, slippage) may occur. 

Radial Variations in Bed Structure As discussed in Section 4.10.6.4, radial variations in a packed bed occur in shallow reactors with a low ratio of the tube to particle diameter (<10). For lower values, a non-uniform radial velodty profile is induced and significant undesirable wall effects (bypassing, slippage) may occur. Consequently, the criterion for negligible influence of radial variations in bed structure is ... 

FIGURE 2.22 Flow profiles in tubes and packed columns. Segment A, laminar flow r = tube radius, Fj = stream path velocity at radial position r, Fmax = maximum flow velocity at tube center. Most open tubular columns operate with this profile. Segment B, turbulent flow 1 = laminar sublayer, 2 = buffer layer. Segment C, plug flow. Segment D, flow in a packed column. Effect is more pronounced with smaller tube diameter -particle size ratios. 

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