Mean residence time model

The results of Example 5.2 apply to a reactor with a fixed reaction time, i or thatch- Equation (5.5) shows that the optimal temperature in a CSTR decreases as the mean residence time increases. This is also true for a PFR or a batch reactor. There is no interior optimum with respect to reaction time for a single, reversible reaction. When Ef < Ef, the best yield is obtained in a large reactor operating at low temperature. Obviously, the kinetic model ceases to apply when the reactants freeze. More realistically, capital and operating costs impose constraints on the design. 

Consider the scaleup of a small, tubular reactor in which diffusion of both mass and heat is important. As a practical matter, the same fluid, the same inlet temperature, and the same mean residence time will be used in the small and large reactors. Substitute fluids and cold-flow models are sometimes used to study the fluid mechanics of a reactor, but not the kinetics of the reaction. 

Real reactors can have 0 < cr < 1, and a model that reflects this possibility consists of a stirred tank in series with a piston flow reactor as indicated in Figure 15.1(a). Other than the mean residence time itself, the model contains only one adjustable parameter. This parameter is called the fractional tubularity, Xp, and is the fraction of the system volume that is occupied by the piston flow element. Figure 15.1(b) shows the washout function for the fractional tubularity model. Its equation is... 

This equation can be fit to experimental data in several ways. The model exhibits a sharp first appearance time, tf st = rpt, which corresponds to the fastest material moving through the system. The mean residence time is found using Equation (15.13), and Xp = tf,rsi/1 is found by observing the time when the experimental washout function first drops below 1.0. It can also be fit from the slope of a plot of In W versus t. This should give a straight line (for t > tfirst) with slope = 1/(F— tfirst)- Another approach is to calculate the dimen-... 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

In the case of a one-compartment open model with single-dose intravenous administration, the mean residence time is simply the inverse of the elimination transfer constant kp, since according to the above definition we obtain ... 

The mean residence time MRT can thus be defined as the time it takes for a single intravenous dose to be reduced to 36.8% in a one-compartment open model. This follows from the property derived above in eq. (39.105) and from eq. (39.5) ... 

For exothermic, reversible reactions, the existence of a locus of maximum rates, as shown in Section 5.3.4, and illustrated in Figures 5.2(a) and 18.3, introduces the opportunity to optimize (minimize) the reactor volume or mean residence time for a specified throughput and fractional conversion of reactant. This is done by choice of an appropriate T (for a CSTR) or T profile (for a PFR) so that the rate is a maximum at each point. The mode of operation (e.g., adiabatic operation for a PFR) may not allow a faithful interpretation of this requirement. For illustration, we consider the optimization of both a CSTR and a PFR for the model reaction... 

Asif et al. (1991) studied distributor effects in liquid-fluidized beds of low-density particles by measuring RTDs of the system by pulse injection of methylene blue. If PF leads into and follows the fluidized bed with a total time delay of 10 s, use the following data to calculate the mean-residence time and variance of a fluid element, and find N for the US model. 

The PFR model is based on turbulent pipe flow in the limit where axial dispersion can be assumed to be negligible (see Fig. 1.1). The mean residence time rpfr in a PFR depends only on the mean axial fluid velocity (U-) and the length of the reactor Lpfr ... 

Primary outputs are produced essentially by sedimentation and (to a much lower extent) by emissions in the atmosphere. The steady state models proposed for seawater are essentially of two types box models and tube models. In box models, oceans are visualized as neighboring interconnected boxes. Mass transfer between these boxes depends on the mean residence time in each box. The difference between mean residence times in two neighboring boxes determines the rate of flux of matter from one to the other. The box model is particularly efficient when the time of residence is derived through the chronological properties of first-order decay reactions in radiogenic isotopes. For instance, figure 8.39 shows the box model of Broecker et al. (1961), based on The ratio, normal-... 

A number of publications (6-10) have demonstrated that the size separation mechanism In HDC can be described by the parallel capillary model for the bed Interstices. The relevant expression for the separation factor, Rj., (ratio of eluant tracer to particle mean residence times) Is given by. 

In the case of porous HDC, as Indicated, one needs to account for both HDC, pore partitioning, and hindered diffusion processes. A model should also have as asymptotes the mean residence time behavior given by Equations 1 to 3 for a nonporous system and Equation 4 for a purely flow-through porous system. 

A model based on the assumption that a metabolite is present within a single compartment with defined rate constants for absorption and elimination of the metabolite. The rate of appearance of a tracee and the infusion of tracer are assumed to take place in a single pool that is instantly well-mixed. Wolfe has described in detail how the constant tracer infusion method allows one to calculate half-life, pool size, turnover time, mean residence time, and clearance time. 

The first term of Eq. (140) represents flow through the active portion of the vessel with a mean residence time of ta = Vt/vi. The second term of Eq. (140) represents the fluid which is bypassing the vessel. This model... 

Level B Correlation A predictive mathematical model for the relationship between summary parameters that characterize the in vitro and in vivo time courses, e.g., models that relate the mean in vitro dissolution time to the mean in vivo dissolution time, the mean in vitro dissolution time to the mean residence time in vivo, or the in vitro dissolution rate constant to the absorption rate constant. 

Figure 6.4 shows how the extent of reaction depends on the mean residence time (l//cf) for the simple first-order reaction model, with... 

In the spirit of the box model approach, of the several residence times of geophysical interest two now seem to be fairly well established— the mean residence time of a trace substance in the lower stratosphere before entering the troposphere and the mean residence time of C02 in... 

Here it was assumed that n=3. From bqs. 14 and IS, we calculate that o2(r0)=0.333. and K/vl. 0 115. Be tween the two predictions is a composite curve. This is an interpolation predicting what the retention time distribution should be for the constant-level skim tank based on the two models. According to this curve, the peak concentration is predicted to occur at t-O. Stp, and the time span at one half the peak concentration is t. The peak concentration should be about 0.9 (/ ). The variance. aJ, of this vessel, should he 0.333 the mean residence time is 0.78fp. The actual retention time distribution for this vessel is plotted in Fig. 6. It can he seen that the peak concentration actually occurs at /=O.35r0. From Eq. 8. T=54.9 minutes (1.0lro) for this distribution and we can calculate that 02( d)sO.3O3. Thus, in this vessel the actual constants are rt 3.29 and A/vT.=0.106, using Eqs. 14 and IS. 

E. Nakashima and L. Z. Benet. General treatment of mean residence time, clearance, and volume parameters in linear mammillary models with elimination from any compartment. J. Pharm. Biopharm. 16 475-492, 1988. 

This is the model for the overall residence time distribution of the particles in the impinging stream contactor under consideration. The model contains several parameters related to equipment structure and operating conditions, i.e. the mean residence times in the four sub-spaces, 7ac, 7im, 7fai and cs. Among the four parameters, the mean residence time in the impingement zone, t m, and that in the collision-slipping region, fcs, are symmetrical parameters, which have the same influence on the overall residence time distribution. It can be seen from Eq. (3.27) or... 

Comparison between Experimental Results and Model Predictions. As will be shown later, the important parameter e which represents the mechanism of radical entry into the micelles and particles in the water phase does not affect the steady-state values of monomer conversion and the number of polymer particles when the first reactor is operated at comparatively shorter or longer mean residence times, while the transient kinetic behavior at the start of polymerization or the steady-state values of monomer conversion and particle number at intermediate value of mean residence time depend on the form of e. However, the form of e influences significantly the polydispersity index M /M of the polymers produced at steady state. It is, therefore, preferable to determine the form of e from the examination of the experimental values of Mw/Mn The effect of radical capture mechanism on the value of M /M can be predicted theoretically as shown in Table II, provided that the polymers produced by chain transfer reaction to monomer molecules can be neglected compared to those formed by mutual termination. Degraff and Poehlein(2) reported that experimental values of M /M were between 2 and 3, rather close to 2, as shown in Figure 2. Comparing their experimental values with the theoretical values in Table II, it seems that the radicals in the water phase are not captured in proportion to the surface area of a micelle and a particle but are captured rather in proportion to the first power of the diameters of a micelle and a particle or less than the first power. This indicates that the form of e would be Case A or Case B. In this discussion, therefore, Case A will be used as the form of e for simplicity. 

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