Distributions of contact times

In 1951,Danckwerts [4] proposed the surface renewal model as an extension ofthe penetration model. Instead of assuming a fixed contact time for all fluid elements, Danckwerts assumed a wide distribution of contact time, from zero to infinity, and supposed that the chance of an element ofthe surface being replaced with fresh liquid was independent of the length of time for which it has been exposed. Then, it was shown, theoretically, that the averaged mass transfer coefficient at the interface is given as... 

Danckwerts8 reasoned that some surfaces may remain in contact longer than other surfaces and generalized the penetration theory by introducing a probability function which expresses the fact that there exists a distribution of contact times lying between zero and infinity. In the resulting expression obtained for Na, the term 2 /D/nt is replaced by yJWs, where s is the fractional renewal rate. Since the unknown t is replaced by the unknown s, the model is not suitable for the prediction of mass transfer coefficients. 

Further relaxation of the assumptions of the film and penetration theories was suggested by Danckweits who viewed the process as one of transient one-dimensional diffusion to packets or elements of fluid that reside at the phaM interface for varying periods of time. Therefore, the model is that of the penetration theory with a distribution of contact times. The surface age distribution (r) is defined such that is the fraction of surface that has resided at the interface for a time between I and i + dt. The mass transfer flux for the entire surface is obtained by integration of the instantaneous flux over all exposure times ... 

Various distributions of contact times of the solid particle with the wall could be considered. The exact distribution can be found from analysis of the experimental data. Considering a distribution of contact times of the solid particles with the wall as... 

Example 6-6 Data taken on the bombardment of the wall of a Plexiglas tube show that the distribution of contact times can be represented as ( t) = se with s = 0.1 sec l. For 150-jum-diameter glass beads, estimate the transfer coefficient h for electrostatic charge transfer. 

The distribution of contact times in a catalyst bed is governed by localized area of pressure drop. Inaccurate calculations of the pressure drop through the ammonia converter can result in severe bottlenecking of the whole plant. Since the literature dealing with this matter is often vague and unreliable, it is appropriate at this point to discuss briefly this subject and also to give a method for calculation. 

Walter et al. studied the flow distribution in simple multichannel geometries by means of the finite-element method [112]. In order to reduce the computational effort, a 2-D model was set up to mimic the 3-D multichannel geometry. Even at a comparatively small Reynolds number of 30 they found recirculation zones in the flow distribution chamber and corresponding deviations from the mean flow rate inside the channels of about 20%. They also investigated the influence of contact time variation on a simple two-step reaction. 

Fig. 3. Distribution of butene as a function of contact time. 2-ButanoI over AljO, (from isopropoxide) at 350°.

Figure 6. Distribution coefficients as functions of contact time and surface to mass ratios for Cs, Sr, and Am on granite (25°C, Aq os)...

Water Treatment. The source of water for this experiment was a pilot plant located on the Seine River upstream from Paris, France (Figure 1). The pilot plant uses an upflow solids contact clarifier (Pulsator, Degremont, Rueil Malmaison, France) followed by rapid sand filtration (RSF). The filtered water is then distributed over four treatment lines to evaluate the efficiency of various ozone-GAC combinations (ozonation rates of 1 or 5 ppm O3 and 10-30 min of contact time). The GAC used in this study was Calgon F-400 (Calgon Corp.). Disinfection by chlorine or chlorine dioxide completed the process. In this chapter, line 3 treatment was not considered a complete treatment for the water supply. This line was studied to evaluate the efficiency of a high ozonation rate. 

The scheme (Figure 3.5) of the transformation of this P - acetic acid mixture over HMFI at 553 K was established from the effect of conversion (or of contact time) on product distribution.[69] At short contact times (i.e. at low conversions), PA was practically the only reaction product. However, there was also formation of a very small amount of o-HAP (Figure 3.6). This means that O acetylation is much faster than C acetylation (ca. 20 times) and that the latter reaction leads to the ortho isomer only. Because of the high rate of O acetylation, thermodynamic equilibrium between P, acetic acid and PA was established at relatively short contact times. At high conversions, the formation of o-HAP involved the participation of PA as demonstrated by the decrease in PA yield and the apparent secondary mode of o-HAP formation (Figure 3.6). This mode of o-HAP formation from PA is mainly intermolecular involving the acetylation of P by PA. The intramolecular transformation of PA into o-HAP is much slower as shown by comparing the transformations of pure PA and of an equimolar mixture of PA and P.[69] The formation of small amounts of p-HAP would result mainly from the hydrolysis of... 

Glasser, D., Katz, S., and Shinnar, R., the measurement and interpretation of contact time distributions for catalytic reactor characterization./nd. Eng. Chem. Fundament. 12,165 (1973). 

Figure 3.92. Distribution of radioactive cholesterol at the boundary between a radioactive and a non-radioactive layer, as a function of contact time o initial distribution, distribution after 15.5 h D° = 0.61 x 10" s" ) A, distribution after 41.5 h, r = 40 mN...

Reaction mechanism and the structure of active site, over Cu-ZnO based catalyst Many experimental results suggest methanol is produced not from CO, but from CO2 directly. As mentioned above, methanol yield from H2/CO2 is higher than that from H2/CO at a wide range of reaction conditions.[5,6,7,20] A study of influence of contact time on product distribution suggested methanol was a primary product[21]. Fujita et al.[7] and Koeppel et al.[22] showed methanol and CO were produced... 

In the calculations, contact time is represented by which is the time needed for 10% of the water to pass through the basin. In other words, describes the time, in minutes, that 90% of the water remains in the basin. For the distribution pipe, contact time is 100% of the time that water remains in the pipe. 

The catalyst, 50% M/H-ZSM-5 and 50% alumina, was supplied by INTEVEP S.A. as 1/16 inch diameter cylinders. Its catalytic properties (activity, yield and selectivity) was studied in an isothermal fixed bed reactor operated under hydrogen atmosphere keeping the ratio of H/n-C8 always equal to 42.06. Deactivation tests were done under hydrogen atmosphere and at the following reaction conditions WWH = 2.87 h, T = 280°C, P = 200 psig., until a stable activity level or a complete deactivation was reached. For pure n-octane transformation over a stabilized catalyst, the change in products distribution with contact time, fi om 0.2 to 2 h, and with reaction temperatures, fi om 200 to 470°C, was also studied [9]. 

The continuous phase is well mixed but there is a dispersed phase. The particles in the dispersed phase behave as PFRs. They are in contact with the continuous phase but are isolated from each other and have an exponential distribution of residence times. This case is treated in Examples 11.17 and 15.12. 

This assumes that all chemical species have the same residence time distribution, and is very convenient to compute the reaction paths for different contacting patterns. Matsuyama and Miyauchi [16] have also considered some aspects of this. An important conclusion of Wei [15] is that for a reactor with distribution of residence times, all reactions are slowed down in comparison with those in a plug flow reactor, but the faster reactions are slowed down a great deal more than the slower ones. Consequently, the occurrence of distribution of residence times makes all reaction rates of the characteristic species nearly equal. That is, the differences between the various reaction rates are decreased, thereby decreasing the selectivity. This is similar to the diffusion effects considered in Chapter 3. 

Figure 8.10 Variation of product distribution with contact time for the reaction of methanol to hydrocarbons over zeolite H-ZSM-5. [Figure reproduced from reference 75 with permission. Copyright 1977 Elsevier.]...

Porter,K. "The effect of contact time distribution on gas absorption with chemical reaction".Trans.Instn.Chem.Enqrs. 44 (1966) T25. 

Other questions which are presently under study are (i) the effect of contacts, (ii) the physical origin of a quadratic co-dependence of a, (iii) the lower limit to the distribution of relaxation times, (iv) the low frequency tail of the lattice vibration absorption, and the a.c. loss mechanisms at high fields (Jonscher (1971, 1972)). Some of these questions will be discussed briefly in the following. 

Assume that a finite system is in contact with a heat bath at constant temperature and driven away from equilibrium by some external time-dependent force. Many nonequilibrium statistical analyses are available for the systems in the vicinity of equilibrium. The only exception is the fluctuation theorems, which are related to the entropy production and valid for systems far away from global equilibrium. The systems that are far from global equilibrium are stochastic in nature with varying spatial and timescales. The fluctuation theorem relates to the probability distributions of the time-averaged irreversible entropy production a. The theorem states that, in systems away from equilibrium over a finite time t, the ratio between the probability that CT takes on a value A and the probability that it takes the opposite value, —A, will be exponential in At. For nonequilibrium system in a finite time, the fluctuation theorem formulates that entropy will flow in a direction opposite to that dictated hy the second law of thermodynamics. Mathematically, the fluctuation theorem is expressed as ... 

Figure 47 Effect of contact time on product distribution during HDS of DBT over M02C/CBC catalyst (653 K 5.0 MPa) ...

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