Turbulent flow through packed beds

In a straight pipe, the transition from streamline to turbulent flow occurs at Re 2300. For flow through packed beds, the transition occurs at a value of Rep of approximately 40. 

We see that Apjl, the frictional pressure drop per unit depth of bed, is made up of two components. The first term on the right-hand-side accounts for viscous (laminar) frictional losses, cc pu. and dominates at low Reynolds numbers. The second term on the right-hand-side accounts for the inertial (turbulent) frictional losses, oc pu2, and dominates at high Reynolds numbers. For further information about flow through packed beds, see Chapter 7 An Introduction to Particle Systems . 

If a population of particles is to be represented by a single number, there are many different measures of central tendency or mean sizes. Those include the median, the mode and many different means arithmetic, geometric, quadratic, cubic, bi-quadratic, harmonic (ref. 1) to name just a few. As to which is to be chosen to represent the population, once again this depends on what property is of importance the real system is in effect to be represented by an artificial mono-sized system of particle size equal to the mean. Thus, for example, in precipitation of fine particles due to turbulence or in total recovery predictions in gas cleaning, a simple analysis may be used to show that the most relevant mean size is the arithmetic mean of the mass distribution (this is the same as the bi-quadratic mean of the number distribution). In flow through packed beds (relevant to powder aeration or de-aeration), it is the arithmetic mean of the surface distribution, which is identical to the harmonic mean of the mass distribution. 

Data acquired by many investigators have shown a close analogy between the rates of heat and mass transfer, not only in the case of packed beds but also in other cases, such as flow through and outside tubes, and flow along flat plates. In such cases, plots of the /-factors for heat and mass transfer against the Reynolds number produce almost identical curves. Consider, for example, the case of turbulent flow through tubes. Since... 

In the less turbulent flow through the straight channel of a monolith, momentum transfer from the fluid to the wall is less effective and in the case of two-phase, countercurrent annular flow, momentum transfer between gas and liquid will also be less than in the interstitial channels of a packed bed. The lower rates of momentum transfer, which is the reason for the higher permeability of monoliths, should in principle improve the possibility for achieving countercurrent flow of gas and liquid at realistic velocities. 

Uniform fluid flow through the bed is desirable for good utilization of the catalyst and control of the process. To avoid channeling, the bed is packed as evenly as possible. A rule of thumb dictates that the reactor to particle diameter ratio should be from five to ten, with the reactor length at least SO-100 times the particle diameter." This ensures that the flow is turbulent, uniform, and approximates plug flow. For most commercial reactors these criteria are met. Only in the narrow>tube reactors found in highly endo- or exothermic processes is there any concern. 

Estimate convective mass-transfer coefficients for the following situations (a) flow paralell to a flat surface, (b) flow past a single sphere, (c) flow normal to a single cylinder, (d) turbulent flow in circular pipes, (e) flow through packed and fluidized beds, and (f) flow through the shell side of a hollow-fiber membrane module. 

This is a very high pressure drop, even though the packed bed is very shallow For comparison purposes, let us estimate the pressure drop for the wetted-wall column of Example 2.12. To estimate the pressure drop in turbulent flow through a circular smooth pipe of diameter D and length Z (Welty et al., 1984),... 

For turbulent flow through a randomly packed bed of monosized spheres of diameter x the equivalent equation is ... 

WH Gauvin, S Katta. Momentum transfer through packed beds of various particles in the turbulent flow regime. AIChE J 19 775 783, 1973. 

At low fluid velocities through packed beds of powders the laminar flow term predominates, whereas at higher velocities both viscous and kinetic effects are important. Er n and Oming [14] found that in the transitional region between laminar and turbulent flow, the equation relating pressure gradient and superflcial fluid velocity uf was ... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

The third and fourth condition are fulfilled by Tarhan [25]. Axial dispersion is fundamentally local backmixing of reactants and products in the axial, or longitudinal direction in the small interstices of the packed bed, which is due to molecular diffusion, convection, and turbulence. Axial dispersion has been shown to be negligible in fixed-bed gas reactors. The fourth condition (no radial dispersion) can be met if the flow pattern through the bed already meets the second condition. If the flow velocity in the axial direction is constant through the entire cross section and if the reactor is well insulated (first condition), there can be no radial dispersion to speak of in gas reactors. Thus, the one-dimensional adiabatic reactor model may be actualized without great difficulties. ... 

The gas flow through tubular reactors is of particular importance because the composition at any point is influenced by the linear velocity of the gas, the size of the reactor and the size of the catalyst particles. When gas flows through a pipe at low linear velocity (low Reynolds number), the radial velocity is not uniform. As the linear velocity increases, turbulence increases and the velocity profile approaches what is called plug flow. However, in a packed bed, plug flow can never be completely attained because of the high voidage near the reactor wall. 

Fig. 7.9 shows the effect ofpH on the measured haematite deposit mass per unit area with time. In these experiments the particulate concentration was 100 mg/kg and the Reynolds number 11,000. The high levels of deposit mass per unit area at a pH of around 6 may also be compared to the data on Fig. 7.10. The pH controls the magnitude and sign of charges on particles and substrate, as shown in the work of Matijevic [1982]. It may be possible with systems having these characteristics that particle deposition could be limited by suitable pH control. It is also interesting that the work of Newson et al [1988] with turbulent flow conditions gave similar results to Kuo and Matijevic [1980] for laminar flow through a packed bed.

Fig. 4.2 — Convective diffusion regimes commonly found in industrial cells, (a) flow through channel formed by two parallel electrode (or by one electrode and a membrane), (b) flow through such a channel but containing a turbulence promoter (e.g. a set of non-conducting bars or a net), (c) fluidised bed electrode, (d) packed bed electrode, (e) rotating cylinder electrode within a concentric tube.

---