Calculation of the residence time

The rate constant of separation consists of two terms, step-atom formation and direct ad-atom transfer 

The pre-exponential factor kg is approximately the oscillation frequency of the atom in the kink site position. An estimated value of 10 can be used. 

At the interface to a liquid phase the transition complex is not only described by the atoms of the solid phase. The moving atom can form complexes with molecules or ions of 

Rate constants of separation and residence times calculated for some cep and hep metals are shown in Table 7.1. 

Comparison of the dynamic of some metals with cep and hep structures rate constants of separation from kink site positions (calculated with the assumption that the activation energy is proportional to the cleavage of one bond nd residence time t 

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Analogous to the above ordinary Michaelis-Menten kinetics, substitution of Equation (11.56) in Equation (11.21) yields a formula which allows straightforward calculation of the residence time required for a specific conversion ... 

The diffusion coefficient was estimated to be 4 x 10 9 m2/s. Experimental values for benzene in faujasites range from 10 10 to 10"13 m2/s, depending on the measurement technique (24, 97). PFG-NMR measurements are the closest to the MD value, which was admitted by the authors to be a crude estimate (mainly on grounds of a short simulation and inflexible molecules). The simulation time was too short to permit a calculation of the residence times of the benzene at either the cation or the window site or inside a particular cage. The cage residence times were estimated to be at least an order of magnitude longer than those for methane in NaY zeolite (43). 

Influence of Residence Time. An approximate calculation of the residence time of the particles in the jet, assuming that they reach the gas velocity instantaneously (which is probably true for the small particles), shows that this time is about 1 msec. Estimating the rate of heating of particles of various sizes shows that particles less than 50m in diameter will reach a temperature of 1000°C. in approximately 1 msec., assuming a gas temperature of 4700°C. —an average heating rate of 10 °C. per sec. These estimates indicate that... 

Equation (7-54) allows calculation of the residence time required to achieve a given conversion or effluent composition. In the case of a network of reactions, knowing the reaction rates as a function of volumetric concentrations allows solution of the set of often nonlinear algebraic material balance equations using an implicit solver such as the multi variable Newton-Raphson method to determine the CSTR effluent concentration as a function of the residence time. As for batch reactors, for a single reaction all compositions can be expressed in terms of a component conversion or volumetric concentration, and Eq. (7-54) then becomes a single nonlinear algebraic equation solved by the Newton-Raphson method (for more details on this method see the relevant section this handbook). 

The derivations presented here support the hypothesis that the concentration level of cb does not affect that of CRmax—in the case of mixed reactions where the participating reactions are of the same order in B. For the calculation of tmax,PFR> Equation 3.262 can be integrated numerically, assuming that the concentration of species B, cb, corresponding to the given concentration of species A, ca, can a priori be calculated from Equation 3.276. Naturally, the same principle can generally be applied for the calculation of the residence time, t, as a function of ca in BRs and PERs. 

Calculation of the induction time is crucial, since gaining a stable and continuous process requires residence times in the mixer < precipitation induction times in order to prevent incrustation in the mixing device. The induction... 

Experiments of propane pyrolysis were carried out using a thin tubular CVD reactor as shown in Fig. 1 [4]. The inner diameter and heating length of the tube were 4.8 mm and 30 cm, respectively. Temperature was around 1000°C. Propane pressure was 0.1-6.7 kPa. Total pressure was 6.7 kPa. Helium was used as carrier gas. The product gas was analyzed by gas chromatography and the carbon deposition rate was calculated from the film thickness measured by electron microscopy. The effects of the residence time and the temperature... 

Calculate (a) the residence time, t, and (b) the space time, r, and (c) explain any difference between the two, for the gas-phase production of C2H4 from C2H6 in a cylindrical PFR of constant diameter, based on the following data and assumptions ... 

In process operations, simultaneous transfer of momentum, heat, and mass occur within the walls of the equipment vessels and exchangers. Transfer processes usually take place with turbulent flow, under forced convection, with or without radiation heat transfer. One of the purposes of engineering science is to provide measurements, interpretations and theories which are useful in the design of equipment and processes, in terms of the residence time required in a given process apparatus. This is why we are concerned here with the coefficients of the governing rate laws that permit such design calculations. 

Figure 32b shows the variation of the residence time between jumps x with increasing temperature. Furthermore, we have calculated the value for the diffusion constant D = r2/2x. One sees that D increases with temperature up to values of D = 7 x 10 7 cm2s-1 for T = 430 K. From the variation of D with temperature, assuming an activation law... 

Measurement of axial mixing in the liquid phase of a fluidized bed is performed by analysis of the residence time distribution of step or pulse signals [55], By plotting the dimensionless E-function of the output signal versus the dimensionless time, the moments of the residence time distribution may be calculated according to Eqs. (7) and (8), the first dimensionless moment /q describing the mean residence time and the second dimensionless moment U2 standing for the variance of the distribution. 

From the point of view of practical application, it is certainly of interest to understand, even primarily, the possible residence time of the particle in the impingement zone by a theoretical analysis. In principle, the total residence time of the particle, fr, can be calculated as the summation of the residence times in all the motion stages ... 

P 62] A Lagrangian particle tracking technique, i.e. the computation of trajectories of massless tracer particles, which allows the computation of interfacial stretching factors, was coupled to CFD simulation [47]. Some calculations concerning the residence time distribution were also performed. A constant, uniform velocity and pressure were applied at the inlet and outlet, respectively. The existence of a fully developed flow without any noticeable effect of the inlet and outlet boundaries was assured by inspection of the computed flow fields obtained in the third mixer segment for all Reynolds numbers under study. 

In all the just-mentioned examples, quantitative prediction and design require the detailed knowledge of the residence time distribution functions. Moreover, in normal operation, the time needed to purge a system, or to switch materials, is also determined by the nature of this function. Therefore the calculation and measurement of RTD functions in processing equipment have an important role in design and operation. 

Fig. 3 Theoretical retention curves for catalysts noncovalently anchored to a functionalized dendritic host with various association constants concentration guest catalyst ([Guest]) in the reactor as function of the residence time (Nr). Calculation parameters [Host] = 3.60 x 10-3 M, [Guest]start = 2.80 x 1CT3 M, retention guest catalyst = 96%, retention host = 100%...

Ways of calculating these parameters are well-known. For example, they are simply related to the coefficients in the Taylor1s Series expansion of the Laplace Transform of the equation which describes the temperature transient without reaction. With each of the six reactor models, an expression for the ratio of the variance of the residence-time distribution to the square of the mean can be derived analytically by finding the Laplace Transform. The results of such an analysis are listed in Table X. 

Equations 5.72 and 5.73 show that if an error is made in the determination of the residence time the same relative error will be found in the calculation of kA by both methods. The errors in kA due to inaccurate conversion measurements depend on the conversion level and are different for the two methods. The significance of accurate conversion measurements is demonstrated in Figure 5.9. This figure presents the relative error in kA caused by a 1% error in the determination of the conversion, so that AC = 0.01. 

Solution The calculations for the method of moments can be carried out using the procedure outlined in the previous illustration. The calculations of mean residence time and Peclet number by the method of Ostergaard and Michelsen are carried out as follows. 

The following treatment considers the effect of the residence time distribution on the size distribution of particles produced in a gas phase reactor. To do this we have to assume that the particles are produced by nucleation, either single point at the inlet of the reactor or multipoint through out the reactor, and particle growth is atom by atom with a growth rate G. Using the residence time distribution, the particle size distribution can be calculated for these two cases of nucleation [33]. 

Now, calculate the length of the separator. First, calculate the height of the liquid from Equation 6.9.5. Use an average of the residence times given by Equation 6.9.6. 

Once the variance of the residence time distribution has been obtained, the axial dispersion coefficient of the tracer g sE is calculated by combining the following equation for the one-dimensional diffusion model (V3) with Eq. (6-28) or Eq. (6-31) ... 

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