Residence time distribution , for

Residence Time Distribution For laminar Newtonian pipe flow, the cumulative residence time distribution F(0) is given by... 

RESIDENCE TIME DISTRIBUTION FOR A LAMINAR FLOW TUBULAR REACTOR... 

Figure 8-23. Residence time distribution for tubuiar iaminar fiow.

Glaser and Lichtenstein (G3) measured the liquid residence-time distribution for cocurrent downward flow of gas and liquid in columns of -in., 2-in., and 1-ft diameter packed with porous or nonporous -pg-in. or -in. cylindrical packings. The fluid media were an aqueous calcium chloride solution and air in one series of experiments and kerosene and hydrogen in another. Pulses of radioactive tracer (carbon-12, phosphorous-32, or rubi-dium-86) were injected outside the column, and the effluent concentration measured by Geiger counter. Axial dispersion was characterized by variability (defined as the standard deviation of residence time divided by the average residence time), and corrections for end effects were included in the analysis. The experiments indicate no effect of bed diameter upon variability. For a packed bed of porous particles, variability was found to consist of three components (1) Variability due to bulk flow through the bed... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

Schoenemann (S4) reported qualitatively that the liquid residence-time distribution for cocurrent upward bubble flow was narrower than that observed in trickle-flow operation. 

FIGURE 15.9 Residence time distribution for laminar flow in a circular tube (a) physical representation b) washout function. 

The limits for part (b) are at the endpoints of a vertical line in Figure 15.14 that corresponds to the residence time distribution for two tanks in series. The maximum mixedness point on this line is 0.287 as calculated in Example 15.14. The complete segregation limit is 0.233 as calculated from Equation (15.48) using/(/) for the tanks-in-series model with N=2 ... 

Two template examples based on a capillary geometry are the plug flow ideal reactor and the non-ideal Poiseuille flow reactor [3]. Because in the plug flow reactor there is a single velocity, v0, with a velocity probability distribution P(v) = v0 16 (v - Vo) the residence time distribution for capillary of length L is the normalized delta function RTD(t) = T 1S(t-1), where x = I/v0. The non-ideal reactor with the para-... 

The residence time distribution for a continuous stirred tank reactor may be represented in terms of the F(t) curve as... 

All the preceding sections were concerned with one-dimensional voidage distribution in the vertical direction. However, maldistribution of solids in the radial direction, generally dilute in the center and dense next to the wall, often causes unfavorable residence time distributions for both the solids and the fluidizing gas, thus resulting in undesirable product distribution. Although it has long been known that in vertical flow of G/S systems solids are preferentially scattered toward the wall, accurate measurement has not been easy. 

In the compound mill, the cylinder is divided into a number of compartments by vertical perforated plates. The material flows axially along the mill and can pass from one compartment to the next only when its size has been reduced to less than that of the perforations in the plate. Each compartment is supplied with balls of a different size. The large balls are at the entry end and thus operate on the feed material, whilst the small balls come into contact with the material immediately before it is discharged. This results in economical operation and the formation of a uniform product. It also gives an improved residence time distribution for the material, since a single stage ball mill approximates closely to a completely mixed system. 

In a final RTD experiment, a sheet of dye was frozen as before and positioned in the feed channel perpendicular to the flight tip. The sheet positioned the dye evenly across the entire cross section. After the dye thawed, the extruder was operated at five rpm in extrusion mode. The experimental and numerical RTDs for this experiment are shown in Fig. 8.12, and they show the characteristic residence-time distribution for a single-screw extruder. The long peak indicates that most of the dye exits at one time. The shallow decay function indicates wall effects pulling the fluid back up the channel of the extruder, while the extended tail describes dye trapped in the Moffat eddies that greatly impede the down-channel movement of the dye at the flight corners. Moffat eddies will be discussed more next. Due to the physical limitations of the process, sampling was stopped before the tail had completely decreased to zero concentration. 

Fig. 16. Dimensionless residence time distributions for specified values of D/uL as predicted by eqn. (71) this applies for small extents of dispersion, DjuL < 0.01.

These relationships are of profound importance for, once a reactor has been described by means of a transfer function, they enable the residence time distribution for that reactor to be chsiracterised in terms of its mean, variance, skewness, etc. Such a characterisation in terms of a few low-order moments is often entirely adequate for the requirements of chemical reaction engineering. 

It will be assumed again that the well-known residence time distribution for the CSTR also holds for the dispersed phase. Then the average concentration of C is... 

For steady state flow in a closed vessel (i.e. one in which fluid enters and leaves solely by plug flow) the residence time distribution for any batch of fluid entering must be the same as that leaving (otherwise accumulation would occur). 

Figure 6.50 presents the cumulative residence time distribution for a tube with a Newtonian model and for a shear thinning fluid with power law indices of 0.5 and 0.1. Plug flow, which represents the worst mixing scenario, is also presented in the graph. A Bingham fluid, with a power law index of 0, would result in plug flow. 

Cumulative residence time distribution for a stirring tank with perfect mixing. 

Residence Time Distribution for Guided Flow in Channels... 

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