Time Distributions

From Eq. [11.41] one sees that it is acceptable to equate the longer correlation time with that of the overall rota-tional motion. However, the shorter observed coirelation time is not strictly eqnal to the coirelatlon time of the segmental motion. Only when 8f 6p is 8s = Br, the actual correlation time of the fast motion. 

In these expressinais 9 is the central value, o is the staodsid deviation of the Gaussian, and r is the fiill width at half-maximum of the Loientzian. 

Suppose that the anisotropy decay is described by a single-modal distribution, with a single mean value That part of the anisotropy which diqilays a condation time 8 is given by 

By analogy with a multimodal lifetime distribatioo, it is possible to describe the anisoltrqiy decay a multimodal 

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At higher flow rates the dispersion of the tracer and the lower limit for time resolution of the concentration versus time distribution limits the accuracy to better than dt2%. 

Finally, tlie ability to optically address single molecules is enabling some beautiful experiments in quantum optics. The non-Poissonian photon arrival time distributions expected tlieoretically for single molecules have been observed directly, botli antibunching at short times [112] and bunching on longer time scales [6, 112 and 113]. The fluorescence excitation spectra of single molecules bound to spherical microcavities have been examined as a probe... 

Fig. 7. Residence time distributions where U = velocity, V = reactor volume, t = time, = UtjV, Cj = tracer concentration to initial concentration and Q = reactor volume (a) output responses to step changes (b) output responses to pulse inputs.

Fig. 8. Theoretical residence time distributions A, combustor style approach to plug flow B, turbulent bed (100% backmixed).

Preferential Removal of Crystals. Crystal size distributions produced ia a perfectiy mixed continuous crystallizer are highly constraiaed the form of the CSD ia such systems is determined entirely by the residence time distribution of a perfectly mixed crystallizer. Greater flexibiUty can be obtained through iatroduction of selective removal devices that alter the residence time distribution of materials flowing from the crystallizer. The... 

Variations may be narrowed by various devices, but never eliminated completely. Figure 12 depicts relative particle residence time distributions among four dryers. 

F Cumulative residence time distribution Dimensionless Dimensionless... 

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

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

Economic Pipe Diameter, Laminar Flow Pipehnes for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated oy operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid.. Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtouiau fluids, see SkeUand Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). 

Continuous stirred tank reactor Dispersion coefficient Effective diffusivity Knudsen diffusivity Residence time distribution Normalized residence time distribution... 

FIG. 7-3 Concentration profiles in fiatch and continuous flow a) fiatch time profile, (h) semifiatcli time profile, (c) five-stage distance profile, (d) tubular flow distance profile, (e) residence time distributions in single, five-stage, and PFR the shaded area represents the fraction of the feed that has a residence time between the indicated abscissas. 

Residence Time Distribution (RTD) This is established by injecting a known amount of tracer into the feed stream and monitor-... 

Motionless mixers continuously interchange fluid elements between the walls and the center of the conduit, thereby providing enhanced heat transfer and relatively uniform residence times. Distributive mixing is usually excellent however, dispersive mixing may be poor, especially when viscosity ratios are high,... 

To measure a residence-time distribution, a pulse of tagged feed is inserted into a continuous mill and the effluent is sampled on a schedule. If it is a dry miU, a soluble tracer such as salt or dye may be used and the samples analyzed conductimetricaUy or colorimetricaUy. If it is a wet mill, the tracer must be a solid of similar density to the ore. Materials hke copper concentrate, chrome brick, or barites have been used as tracers and analyzed by X-ray fluorescence. To plot results in log-normal coordinates, the concentration data must first be normalized from the form of Fig. 20-15 to the form of cumulative percent discharged, as in Fig. 20-16. For this, one must either know the total amount of pulse fed or determine it by a simple numerical integration... 

Solution for Continuous Mill In the method of Mori (op. cit.) the residence-time distribution is broken up into a number of segments, and the batch-grinding equation is applied to each of them. The resulting size distribution at the miU discharge is... 

The annular gap mill shown in Fig. 20-36 is avariation of the bead mill. It has a high-energy input as shown in Fig. 20-37. It may be lined with polyurethane and operated in multipass mode to narrow the residence-time distribution and to aid cooling. 

The vertical vibratoiy mill has good wear values and a low-noise output. It has an unfavorable residence-time distribution, since in continuous operation it behaves like a well-stirred vessel. Tube mills are better for continuous operation. The mill volume of the vertical mill cannot be arbitrarily scaled up because the static load of the upper media, especially with steel beads, prevents thorough energy introduction into the lower layers. Larger throughputs can therefore only be obtained by using more mill troughs, as in tube mills. 

The vibratoiy-tube mill is also suited to wet milhng. In fine wet milling this narrow residence time distribution lends itself to a simple open circuit with a small throughput. But for tasks of grinding to colloid-size range, the stirred media mill has the advantage. 

Ultrafine grinding is carried out batchwise in vibratoiy or ball mills, either diy or wet. The purpose of batch operation is to avoid the residence time distribution which would pass less-ground material through a continuous mill. The energy input is 20-30 times greater than for standard grinding, with inputs of 1300-1600 kWh/ton compared to 40-60. Jet milling is also used, followed by air classification, which can reduce top size Below 8 [Lm. 

Residence-time distribution is important in continuous mills. Further data are given in the above references. 

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