Fixed beds Peclet number

It is interesting to check the Peclet number of the fixed bed. The Reynolds number is 0.154, and for this low value, the most appropriate correlation is that of Chung (eq. (3.314)). The resulting particle Peclet number is 0.39 and thus, the bed Peclet number is 151.98, which is fairly high, and we can assume that the plug-flow condition is assured. 

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 ... 

When a number of competing reactions are involved in a process, and/or when the desired product is obtained at an intermediate stage of a reaction, it is important to keep the residence-time distribution in a reactor as narrow as possible. Usually, a broadening of the residence-time distribution results in a decrease in selectivity for the desired product. Hence, in addition to the pressure drop, the width of the residence-time distribution is an important figure characterizing the performance of a reactor. In order to estimate the axial dispersion in the fixed-bed reactor, the model of Doraiswamy and Sharma was used [117]. This model proposes a relationship between the dispersive Peclet number ... 

The figure shows the ratio of the widths of initially delta-like concentration tracers at the reactor exits as a function of the micro-channel Peclet number for different values of the porosity. Taking a value of = 0.4 as standard, it becomes apparent that the dispersion in the micro-channel reactor is smaller than that in the fixed-bed reactor in a Peclet number range from 3 to 100. Minimum dispersion is achieved at a Peclet number of about 14, where the tracer width in the micro-channel reactor is reduced by about 40% compared with its fixed-bed counterpart. Hence the conclusion may be drawn that micro-channel reactors bear the potential of a narrower residence time than fixed-bed reactors, where again it should be stressed that reactors with equivalent chemical conversion were chosen for the comparison. 

Figure 1.16 Ratio of the width of concentration tracers at the exit of the micro-channel reactor to the corresponding quantity in the fixed-bed reactor as a function of the micro-channel Peclet number udf /D, as obtained in [114. ...

Multiplying this number with the term ZJd, where Z is the fixed-bed length, we obtain the vessel Peclet number. A high vessel Peclet number means better flow quality, thus closer to ideal flow. Typically, if this number is higher than about 100, the flow is considered to be ideal (plug flow). 

Liquid-solid fixed beds In the related literature, there are correlations for the evaluation of the particle Peclet number (Pefi for materials that are frequently used in adsorption and... 

Liquid-solid fixed beds. According to Gunn (1968), for random beds of spheres (e = 0.4), the radial Peclet numbers are from about 10-40 for 0.08 < Rep < 1000. 

Here, it has to be noted that for calculating the Peclet number in fixed beds, the actual velocity has to be used, i.e. the interstitial velocity, which influences the degree of mixing. In slurry bubble column reactors, the real velocity of the fluid is the bubble velocity, which is much higher than the gas superficial velocity. The mean bubble rise velocity for a batch liquid is (eq (3.201))... 

A demonstration of this approach has been reported to evaluate the ability of a lattice-Boltzmann code to predict both spatially resolved flow fields and MR propagators characterizing flow through random packings of spheres (model fixed beds) for flows defined by Peclet (Pe) and Reynolds numbers in the range 182 < Pc <3 50 and 0.4 < Re <0.77 (85). Excellent agreement was found between the numerical predictions and experimental measurements. Current interest in this field addresses the validation and development of numerical codes predicting flows at Reynolds numbers more appropriate to real catalytic reactors. 

Figure 17.1. Residence time distributions of some commercial and pilot fixed bed reactors. The variance, the equivalent number of CSTR stages, and the Peclet number are given for each.

The membrane reactor has an additional degree of freedom allowing to optimize the amount of the product removed. This can be conveniently described by a dimensionless Peclet number which relates convective flow through the reactor to transport through the membrane [50]. For each Da number there exists an optimal Pe number maximizing the conversion. For a given membrane material the Pe number is directly related to the membrane thickness. Curve (b) in Fig. 12.12 represents the theoretical behavior of a membrane reactor optimized with respect to membrane thickness. For low Da numbers the membrane should be very thick in order to keep the reactants in the reactor, and in this respect the membrane reactor is identical to the conventional fixed-bed reactor. In contrast, for high Da numbers the membrane reactor should possess a very thin wall for... 

In the vast majority of experimental studies, the backmixing characteristics of a flowing phase are examined using a -pulse tracer input. For the fixed-bed systems shown in Fig. 3-2, if a perfect pulse input is used, then, as shown by Levenspiel,5 6 the axial dispersion coefficient or the Peclet number can be obtained from the variance of the RTD curve. For example, for a closed system and large extent of dispersion, the variance, it, is related to the Peclet number by the equation... 

The residence-time distribution in the liquid phase of a cocurrent-upflow fixed-bed column was measured at two different flow rates. The column diameter was 5.1 cm and the packing diameter was 0.38 cm. The bed void fraction was 0.354 and the mass flow rate was 50.4 g s l. The RTD data at two axial positions (which were 91 cm apart in Run 1 and 152 cm apart in Run 2) are summarized in Table 3-2. Using the method of moments, estimate the mean residence time and the Peclet number for these two runs. If one assumes that the backmixing characteristics are independent of the distance between two measuring points, what is the effect of gas flow rate on the mean residence time of liquid and the Peclet number Hovv does the measured and the predicted RTD at the downstream positions compare in both cases ... 

Chung and Wen (1968) and Wen and Fan (1975) have proposed a dimensionless equation using the dependency of the dispersion coefficient on the (particle) Reynolds number Re (Eq. 6.169) for fixed and expanded beds. It is an empirical correlation based on published experimental data and correlations from other authors that covers a wide range of Re. Owing to two different definitions of the Reynolds number, the actual appearance varies in the literature. Since the particle diameter dp, is the characteristic value of the packing, Eq. 6.168 based on the (particle) Peclet number Pe (Eq. 6.170) is used here ... 

The Peclet number for dispersion is defined as Pe = v dpl B, which is dimensionless, and vjEg = V = the interstitial velocity. A convenient way to comprehend the data for fixed beds is via Figure... 

Radial dispersion of mass and heat in fixed bed gas-solid catalytic reactors is usually expressed by radial Peclet number for mass and heat transport. In many cases radial dispersion is negligible if the reactor is adiabatic because there is then no driving force for long range gradients to exist in the radial direction. For non-adiabatic reactors, the heat transfer coeflScient at the wall between the reaction mixture and the cooling medium needs also to be specified. 

Flow of liquids or gases through fixed beds is very important in chemical reaction engineering, since many commercially important processes involve reactors that contain beds of catalyst used to promote a desired reaction. The axial dispersion model has been used extensively to model these flows, even though two phases, fluid and solid, are present. Such a pseudo-homogeneous model assumes the same form we have described in the preceding section if the Peclet number is based on particle dimension and the interstitial fluid velocity is used. In this event... 

The following two models are frequently used to account for partial macromixing the dispersion model and the tanks-in-series model. In the dispersion model, deviation from plug flow is expressed in terms of a dispersion or effective axial diffusion coefficient. This model was anticipated in Chapter 12, and the governing equations for mass and heat are listed in Table 12.2 of that chapter. The derivation is similar to that for plug flow except that now a term is included for diffusive flow in addition to that for bulk flow. This term appears as -D ( d[A]/d ), where is the effective axial diffusion coefficient. When the equation is nondimensionalized, the diffusion coefficient appears as part of the Peclet number defined as = itd/D. A number of correlations for predicting the Peclet number for both liquids and gases in fixed and fluidized beds are available and have been reviewed by Wen and Fan (1975). 

The equation with axial mixing given in Table 12.2 for a fixed-bed reactor also holds for a homogeneous reactor, except that the effective diffusion coefficient is replaced by the real Df. This can be recast as /Pe )(d a/d ) da/dz) — (Z,/[v4]ou)(-rvA) = where Pe i is now the axial Peclet number defined as uL D(. This equation has been solved (see Danckwerts, 1953 Wehner and Wilhelm, 1956) to obtain an expression for concentration as a... 

To highlight the benefits from microstructuring in reducing dispersion, equivalent fixed bed and microreactor/monolith were compared in terms of a dispersion ratio (ratio of the widths of initially delta-like concentration tracers at the reactor exit) [88, 89]. In terms of the Peclet number (with Dax as the axial dispersion), an expression for fixed bed reactor of the type... 

The Peclet number for the process and the various aspect ratios of the fixed-bed reactor determine the impact of the dispersion terms. The mass Peclet number quantifies the ratio of bulk mass transport to diffusive mass transport. We define the mass Peclet number as... 

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