Dispersion Bodenstein number

The inverse of the Bodenstein number is eD i/u dp, sometimes referred to as the intensity of dispersion. Himmelblau and Bischoff [5], Levenspiel [3], and Wen and Fan [6] have derived correlations of the Peclet number versus Reynolds number. Wen and Fan [6] have summarized the correlations for straight pipes, fixed and fluidized beds, and bubble towers. The correlations involve the following dimensionless groups ... 

Naturally, there are two more Peclet numbers defined for the transverse direction dispersions. In these ranges of Reynolds number, the Peclet number for transverse mass transfer is 11, but the Peclet number for transverse heat transfer is not well agreed upon (121, 122). None of these dispersions numbers is known in the metal screen bed. A special problem is created in the monolith where transverse dispersion of mass must be zero, and the parallel dispersion of mass can be estimated by the Taylor axial dispersion theory (123). The dispersion of heat would depend principally on the properties of the monolith substrate. Often, these Peclet numbers for individual pellets are replaced by the Bodenstein numbers for the entire bed... 

In order to evaluate the efficacy of the expanded bed technique the plate height (HETP), plate number (N), resolution (Rs), Bodenstein number (Bo), particle Peclet number (Pep) and axial dispersion coefficient (DJ have been calculated and compared with the corresponding values of a traditional HPLC column. N can be expressed by... 

Other factors limiting the overall rate can be external or internal mass transfer, or axial dispersion in a fixed-bed reactor. Pertinent dimensionless numbers are the Biot number Bi, the Damkohler number of the second kind Dan, or the Bodenstein number Bo (Eqs. (5.46)—(5.48)]. 

Characterizing the distribution according to the dispersion model yields a dimensionless number describing the degree of axial mixing within the bed. The Bodenstein number Bo relates convective transport of liquid to dispersion according to Eq. (9). 

In Fig. 5, the liquid phase axial dispersion coefficient Daxi and the Bodenstein number Bo calculated from this relationship according to Eqs. (9), (10) and (13) are plotted for a range of linear velocities used in fluidized bed adsorption. The physical parameters of the commercial Streamline SP adsorbents (average particle size 247 pm, average particle density 1143 kg/m3, terminal settling velocity 0.0044 m/s, n = 4.7 as described by Chang and Chase [37]) were... 

Axial Convective Diffusion. The variation in width, length and direction of individual channels formed by the interstices of the packing give rise to a dispersion which can be characterized by the dimensionless Bodenstein number. Bo, which is a similar number as the Peclet number but with the particle diameter as characteristic dimension... 

Table IV presents some data on liquid residence time distributions measured under conditions of hydrocracking in trickle flow. It can be seen that bed dilution with fine inert particles results in a considerable improvement in the plug-flow character of the reactor, which supports the idea that the dispersion is largely determined by the packing of fine particles. Since in the range of Re numbers of interest the Bodenstein number is approximately a constant (see Figure 4), the Peclet numbers for beds of equal length should be inversely proportional to the particle diameter. Dilution of the 1.5 mm particles with 0.2 mm particles should raise Pe by a factor of about 7, which is approximately in line with the data in Table IV.

The longitudinal dispersion for flow through a packed bed is correlated with the dimensionless axial Bodenstein number Bo, defined as Bo = dpu/Dapj. At the low linear velocities typical for the operation of the LFR, Bo tends to approach a constant value of approximately 0.4, as found by Gierman [12]. Hence, Eq. (10) can be written as... 

Tanks-in-Series Model Versus Dispersion Model. We have seen that we can apply both of these one-parameter models to tubular reactors using the variance of the RTD. For first-order reactions the two models can be applied with equal ease. However, the tanks-in-series model is mathematically easier to use to obtain the effluent concentration and conversion for reaction orders other than one and for multiple reactions. However, we need to ask what would be the accuracy of using the tanks-in-series model over the dispersion model. These two models are equivalent when the Peclet-Bodenstein number is related to the number of tanks in series, n, by the equation ... 

An alternative model for real flows is the dispersion model with the model parameters Bodenstein number (Bo) and mean residence time t, The Bodenstein number which is defined as Bo = uL/D characterises the degree of backmixing during flow. The parameter D is called the axial dispersion coefficient, u is a velocity and L a length. The RTD of the dispersed plug flow model ranges from PFR at one extreme (Bo = °) to PSR at the other (Bo = 0). The transfer function for the dispersion model with closed-closed boundaries is [10] ... 

Figure 6 shows the example of the reactor dynamics of the dispersion model calculated with a Bodenstein number of 8.8... 

For the SCC of type II an example of a RTD modelled is shown in Figure 7, The model used is the dispersion model (sec Esq. 6). The values of the model parameters determined arc a Bodenstein number of 8.8 and a mean residence time of 0.6 s. It clearly shows that the model for the RTD explains the frequency response measurement up to a frequency of 2 Hz, At the frequency of 2 Hz the signal-to-noise ratio of 100 is reached. Any mixing processes which affect the transfer function above this frequency cannot be identified. 

Dimensionless parameters represent ratios of different mass transport and reaction phenomena. One example is the Bodenstein number (Bo, sometimes called the axial Peclet number), which is the ratio of convection rate to axial dispersion ... 

The dispersion in tubular reactors depends on the flow regime and is characterized by the Bodenstein number, the ratio of the axial diffusion time, tu,ax, in the reactor to the mean fluid residence time, x. 

The axial dispersion in the reactor is often expressed by the axial Peclet number, and the characteristic length, which equals the tube diameter for tubular reactors, and the particle diameter for packed-bed reactors. The Bodenstein number characterizing dispersion in the tubular reactor thus becomes the following ... 

D, which has the same dimension unit as the molecular diffusion coefficient D. Usually is much larger than because it incorporates all effects that may cause deviation from plug flow, such as radial velocity differences, eddies, or vortices. The key parameter determining the width of the RTD is the ratio between the axial dispersion time and the space-time r, which corresponds to the mean residence time in the reactor t at constant fluid density. This ratio is often called Bodenstein number Bo). 

The first term in Eq. (11.38) corresponds to the ratio between space-time and the characteristic axial molecular diffusion time. The molecular diffusion coefficient hes in the order 10 m s for gases and 10 m s for liquids. Typical lengths of MSR are several centimeters and the space-time is in the range of seconds. Therefore, the axial dispersion in microchannels is mainly determined by the second term in Eq. (11.38), where the Bodenstein number can be estimated with Eq. (11.39)... 

FIGURE 22.7 Axial dispersion in the gas phase. Range of Bodenstein numbers as a function of superficial... 

FIGURE 22.8 Axial dispersion in the flowing solids phase. Range of Bodenstein numbers as a function of flowing solids mass flux. Different s3mibols correspond to different values of superficial gas velocity in the range 0-0.130 m/sec. (From Roes, A.W.M. and van Swaaij, W.P.M., Chem. Eng. J., 18, 13, 1979. With permission.)... 

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