Radial void fraction profile

At djdp values below 5, a and b drastically change. From velocity measurements by Bey and Eigenberger [1997] and radial void fraction profiles of de Klerk [2003], Castillo-Araiza and Lopez-Isunda derived values of 270 for a and 2.5 for b. 

Lerou and Froment [l] demonstrated that the structure of the packed bed can be defined by its radial void fraction profile. In order to have adequate and reliable data on void fraction profiles an experimental measuring system was constructed, which is presented here together with suggestions for subsequent analysis. 

For each region a mean value of the void fraction was calculated and a hydraulic radius was defined which was used in a pressure drop correlation. Martin [20] divided the bed into two regions a wall and a bulk region. He calculated for both different flow rates and a different rate of heat transfer. Carbonell [2] also used a two zone model for his analysis of the dispersion phenomena. In more recent work Vortmeyer et al. [5>6] tried to use the complete radial void fraction profile, and so did Chang [3]. They followed the same itinerary outlined by Lerou and Froment [l] and Marivoet et al. [2l]. Starting from the void fraction profile the radial velocity profile is calculated. With both profiles the effective thermal conductivity is established and the temperature and concentration profiles can be calculated by means of a two dimensional pseudo homogeneous model for the reactor. 

Figure 1. Measured radial void fraction profile of a bed packed with small spheres.

To evaluate the importance of the radial variations in the void fraction profiles, simulations with prescibed radial variations in the bed structure were performed for the s3mthesis gas process. The velocity and pressure profiles were significantly altered. Higher void fractions at the wall induces less friction from... 

Two-dimensional homogeneous model with porosity and void fraction profile (2D-HOM-PVP). Radial conversion (a) and temperature (b) profiles at a given distance in the bed and for Rsp =175.2 = 2 1 = 0.07 [Papageorgiou and Froment, 1995]. 

However, bubble nonhomogeneous distribution exists in two-phase shear flow. As yet, the following general trends in void fraction radial profiles are being identified for bubbly upward flow (Zun, 1990) concave profiles (Serizawa et al., 1975) convex profiles (Sekoguchi et al., 1981), and intermediate profiles (Sekoguchi et al., 1981 Zun, 1988). Two theories are currently dominant ... 

Clearly, in the absence of a radial temperature or velocity gradient, no radial mass transfer can exist unless, of course, a reaction occurs at the bed wall. When a system is adiabatic, a radial temperature and concentration gradient cannot exist unless a severe radial velocity variation is encountered (Carberry, 1976). Radial variations in fluid velocity can be due to the nature of flow, e.g. in laminar flow, and in the case of radial variations in void fraction. In general, an average radial velocity independent of radial position can be assumed, except from pathological cases such as in very low Reynolds numbers (laminar flow), where a parabolic profile might be anticipated. 

Here, Dy is an empirical, radial dispersion coefficient and e is the void fraction. The units of diffusivity Dy are square meters per second. The major differences between this model and the convective diffusion equation used in Chapter 8 is that the velocity profile is now assumed to be flat and Dy is an empirically determined parameter instead of a molecular diffusivity. The value of Dy depends on factors such as the ratio of tube to packing diameters, the Reynolds number, and (at least at low Reynolds numbers) the physical properties of the fluid. Ordinarily, the same value for Dy is used for all reactants, finessing the problems of multicomponent diffusion and allowing the use of stoichiometry to eliminate Equation 9.1 for some of the components. Note that Us in Equation 9.1 is the superficial velocity, this being the average velocity that would exist if the tube had no packing. 

The simulation results shown here were obtained with a radial void profile constant over the whole bed length, but the model was also applied to variable radial void profiles. More recently Chigada and Mann [2008] used a model with a 2-dimensional network of voids to assess wall flow patterns for djdp ratios as low as 1.32. For these large void fractions there is a considerable radial flow component and its local value is strongly influenced by the neighboring voids. 

Isothermal Packed Beds. A packed reactor has a velocity profile that is nearly flat and, for the usual case of uniform ain, no concentration gradients will arise unless there is a radial temperature gradient. If there is no reaction exotherm (and if Tj =T aid-> th model of Section 9.1 degenerates to piston flow. This is overly optimistic for a real packed bed, and the axial dispersion model provides a correction. The correction will usually be small. Note that should be replaced by Mj and that the void fraction e should be inserted before the reaction term e.g., ki becomes eki for reactions in a packed bed. Figure 9.7 gives De / (usdp) 2 for moderate values of the particle Reynolds number. This... 

Since with constant system pressure in a self-pressurized reactor the outlet enthalpy remains constant, power changes must result in deviations of the inlet temperature. The void fraction reacts to the power level because all radial deviations from the average enthalpy rise due to the power profile are approximately proportional to the power level. 

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