Concentration, cross-section averaged

The experiments of Dou et al. (1991) also indicate that the heat transfer coefficient varied with radial position across the bed, even for a given cross-sectional-averaged suspension density. Their data, as shown in Fig. 20, clearly indicate that the heat transfer coefficient at the bed wall is significantly higher than that for vertical surfaces at the centerline of the bed, over the entire range of suspension densities tested. Almost certainly, this parametric effect can be attributed to radial variations in local solid concentration which tends to be high at the bed wall and low at the bed centerline. 

The data of Fig. 20 also point out an interesting phenomenon—while the heat transfer coefficients at bed wall and bed centerline both correlate with suspension density, their correlations are quantitatively different. This strongly suggests that the cross-sectional solid concentration is an important, but not primary parameter. Dou et al. speculated that the difference may be attributed to variations in the local solid concentration across the diameter of the fast fluidized bed. They show that when the cross-sectional averaged density is modified by an empirical radial distribution to obtain local suspension densities, the heat transfer coefficient indeed than correlates as a single function with local suspension density. This is shown in Fig. 21 where the two sets of data for different radial positions now correlate as a single function with local mixture density. The conclusion is That the convective heat transfer coefficient for surfaces in a fast fluidized bed is determined primarily by the local two-phase mixture density (solid concentration) at the location of that surface, for any given type of particle. The early observed parametric effects of elevation, gas velocity, solid mass flux, and radial position are all secondary to this primary functional dependence. 

Optical fiber measurement of local solids concentrations of FCC catalyst fluidized in a 9-cm-i.d. column gave the results shown typically in Fig. 26. Analysis of these data showed that the radial voidage profile could be described solely by the cross-section-average voidage e, calculated as shown in Sec. 5.1, and the reduced radial coordinate r/R ... 

Figure 24.6 Temporal evolution of a concentration cloud along a river. The curves show cross-sectional averaged dye concentration measured at six sites in the Waikato River (New Zealand) below an instantaneous transverse line source. From Rutherford (1994).

The first 3D model of FFF was developed in Ref. 2. The 3D diffusion-convection equation was solved with the help of generalized dispersion theory, resulting in the equations for the cross-sectional average concentration of the solute and dispersion coefficients and K2, representing the normalized solute zone velocity and the velocity of the corresponding peak width growth, respectively. Unfortunately, only the steady-state asymptotic values of dispersion coefficients Ki oo) and K2 oo) were determined in Ref. 2, leading to the prediction of the solute peaks much wider than the experimental ones. 

Much effort has been made to study this light-off behavior of catalytic monolith. Oh and Cavendish studied the response of the monolith to a step increase in the feed stream temperature by using a onedimensional two-phase (gas and solid) model. They tracked the cross-sectional average temperature and concentration in each phase and used heat and mass transfer coefficients to describe interphase transport. The results indicated that the light-off occurs at the monolith entrance for a sufficiently high inlet exhaust temperature. For a lower inlet exhaust temperature, the light-off occurs in the downstream section, and the... 

We can now use the Taylor dispersion equation in either of the forms (3-244) or (3-245) to show that the cross-sectionally averaged temperature profile is a Gaussian in the z direction, with the peak concentration remaining at z = 0 (i.e., converting downstream relative to fixed coordinates at the mean velocity U). 

The Gaussian plume illustrated in Figure 6 represents the cross-section of a time-averaged, tracer concentration. That is, if time-series concentration measurements taken at a number of points across the plume were separately averaged over their duration, then one would expect to obtain a Gaussian profile. However, at any one time the instantaneous concentration profile would look very different. Figure 12, a typical instantaneous concentration cross-section, shows the small-scale concentration fluctuations resulting from the interaction of coherent structures... 

In the early 1950s, Taylor recognized that this unsteady-state two-dimensional microscopic mass transfer equation for the tracer s molar density, CA r,z,t), could be simplified at long times. The strategy involves writing an unsteady-state one-dimensional mass balance for the cross-section-averaged concentration of the tracer, defined by... 

It is very common in reactors to have flow predominantly in one direction, say z (e.g., think of tubular reactors). The major gradients then occur in that direction, under isothermal conditions at least. For many cases then, the cross-sectional average values of concentration (or conversion) and temperature might be used in the equations instead of radial point values. The former are obtained from ... 

ATc is the equilibrium constant in concentration units and <(Te>therouit is the total compound cross-section averaged over a thermal population of internal and translational reactant states (reduced mass ft) see equation (1). 

Longitudinal dispersion accounts for the dilution of the cross-sectional average concentration of compounds dissolved in the water due to mixing in the streamwise direction. The longitudinal dispersion coefficient, D,... 

Thus, the solution of Equations 3.71 and 3.72 for the cross-sectional averaged contribution to the concentration profile is given by... 

Figure 3.8 Cross-sectional average concentration (a) and temperature (b) profiles according to a pseudohomogeneous model (full lines two-dimensional dashed lines one-dimensional). After Ref [133] with tj = 160 C and Tw = 100 C.

Figure 3.10 Comparison of the axial temperature (a) and methanol concentration (b) according to several models (cross-sectional averages for 2D models) for a flxed-bed reactor to produce formaldehyde from methanol [135]. (Source Reproduced with kind permission of lordanidis.)...

The nuclear reaction (b) is chiefly induced by thermal neutrons and shows a cross section of 6 10 cm (Maxwell spectrum, T=575 K). Reaction (c) on the other hand is induced by fast neutrons with an effective threshold energy of 2.4 MeV the cross section averaged over the flux of neutrons with energies beyond this threshold value amounts to 8.5 10 cm . Using this data and assuming a lithium concentration of 2 ppm (99.99% Li), which is held constant... 

For illustrative purposes, the following hydrodynamic model and correlations are used in the model calculation of catalytic reactions of propylene ammox-idation to acrylonitrile. These hydrodynamic models and correlations are appropriate for a high-density riser under propylene ammoxidation conditions. Specifically, the axial profiles of cross-sectional averaged solids concentration was obtained by fitting the cluster-based model proposed by Li and Kwauk (1980) with the experimental data from Wei et al. (1998). The axial profile of solids concentration obtained can be expressed as ... 

---