The residence time distribution

From previous chapters, it is clear that the fluid mixing pattern within a reactor of a given size will affect the conversion achieved from that reactor in the case of multiple reaction schemes, the product distribution, and hence yield and selectivity, will also be dependent on mixing and 

E(t) is a probability density function or frequency function and E(f)df is the fraction of material which leaves the system with an age of between t and (t + df) units of time. Since all material must have a residence time between zero and infinity 

E(f) may be integrated to any upper limit t to give the cumulative probability function F(t). This is illustrated in Fig. 2. 

It is clear that, in the particular case illustrated, all material has a residence time of between to and t2 minutes and therefore the mean or average time spent in the system is between these two extremes. For a flow-mixing system in which no adsorption, reaction or change in volume flow rate occurs, this mean residence time is always equal to the volumetric holdup of the system divided by the constant flow rate through the system [1]. Thus, the mean residence time, t, equals V/Q and is seen to be identical 

In many situations, it is convenient to plot a residence time distribution as a function of dimensionless time, 9, that is time normalised with respect to the mean residence time of the system under study. Thus 

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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... 

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. 

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. 

The combination of reac tor elements is facihtated by the concept of transfer functions. By this means the Laplace transform can be found for the overall model, and the residence time distribution can be found after inversion. Finally, the chemical conversion in the model can be developed with the segregation and maximum mixed models. 

FIG. 23-10 Residence time distributions of pilot and commercial reactors. <3 = variance of the residence time distribution, n = number of stirred tanks with the same variance, Pe = Peclet number. 

A distinc tion is to be drawn between situations in which (1) the flow pattern is known in detail, and (2) only the residence time distribution is known or can be calculated from tracer response data. Different networks of reactor elements can have similar RTDs, but fixing the network also fixes the RTD. Accordingly, reaction conversions in a known network will be unique for any form of rate equation, whereas conversions figured when only the RTD is known proceed uniquely only for hnear kinetics, although they can be bracketed in the general case. 

Topics that acquire special importance on the industrial scale are the quality of mixing in tanks and the residence time distribution in vessels where plug flow may be the goal. The information about agitation in tanks described for gas/liquid and slurry reactions is largely apphcable here. The relation between heat transfer and agitation also is discussed elsewhere in this Handbook. Residence time distribution is covered at length under Reactor Efficiency. A special case is that of laminar and related flow distributions characteristic of non-Newtonian fluids, which often occiu s in polymerization reactors. 

In granular catalyst packed reactors, the residence time distribution often is no better than that of a five-stage CSTR battery. 

THE RESIDENCE TIME DISTRIBUTION FUNCTIONS AND THEIR RELATIONSHIPS ... 

Mixing, ideal or complete A state of complete uniformity of composition and temperature in a vessel. In a flow system, the residence time distribution is exponential, ranging from zero to infinity. 

Tracer A substance that is used for measuring the residence time distribution in a vessel. Usually, it is inert and used in small concentrations so as not to change the physical properties of the process fluid appreciably, and analyzable for accuracy. 

In a continuous reaction process, the true residence time of the reaction partners in the reactor plays a major role. It is governed by the residence time distribution characteristic of the reactor, which gives information on backmixing (macromixing) of the throughput. The principal objectives of studies into the macrokinetics of a process are to estimate the coefficients of a mathematical model of the process and to validate the model for adequacy. For this purpose, a pilot plant should provide the following ... 

We will use the NormalDistribution to make the representations of the residence time distribution.The ProbabilityDensityFunction(PDF) ismadeupofthe NormalDistribution... 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

Kramers and Alberda (K20) have reported some data in graphical form for the residence-time distribution of water with countercurrent air flow in a column of 15-cm diameter and 66-cm height packed with 10-mm Raschig rings. It was concluded that axial mixing increased with increasing gas flow rate and decreasing liquid flow rate, and that the results were not adequately represented by the diffusion model. 

Kramers et al. (K21) measured gas residence-time distribution in a mechanically agitated gas-liquid contactor of 0.6-m diameter for various gas velocities and agitator speeds. In the region where agitation has an effect on the gas-liquid interfacial area (cf. the study by Westerterp et al. (W5), Section V,D,1), the residence-time distribution was found to resemble closely that of a perfect mixer. 

From the assumption that liquid volume elements travel as bubble wakes at velocities higher than the average liquid velocity, it follows that the bubble movement must influence the residence-time distribution of the liquid phase. However, no work on this subject has come to the author s attention. 

A further effect given by the system is the narrowing of the residence time distribution of the product in the evaporator, thus reducing the possibility of local or spot overheating and a consequent negative effect on final product quality. 

The residence time distribution for a two-tank system is given by... 

When Equation 9 is used in Equation 8 along with the relationships for the residence time distributions one obtains the following dimensionless particle size distributions for one- and two-tank systems. 

The concept of a well-stirred segregated reactor which also has an exponential residence time distribution function was introduced by Dankwerts (16, 17) and was elaborated upon by Zweitering (18). In a totally segregated, stirred tank reactor, the feed stream is envisioned to enter the reactor in the form of macro-molecular capsules which do not exchange their contents with other capsules in the feed stream or in the reactor volume. The capsules act as batch reactors with reaction times equal to their residence time in the reactor. The reactor product is thus found by calculating the weighted sum of a series of batch reactor products with reaction times from zero to infinity. The weighting factor is determined by the residence time distribution function of the constant flow stirred tank reactor. 

To run the residence time distribution experiments under conditions which would simulate the conditions occurring during chemical reaction, solutions of 15 weight percent and 30 percent polystyrene in benzene as well as pure benzene were used as the fluid medium. The polystyrene used in the RTD experiment was prepared in a batch reactor and had a number average degree of polymerization of 320 and a polydispersity index, DI, of 1.17. 

Constant RTD control can be applied in reverse to startup a vessel while minimizing olf-specification materials. For this form of startup, a near steady state is first achieved with a minimum level of material and thus with minimum throughput. When the product is satisfactory, the operating level is gradually increased by lowering the discharge flow while applying Equation (14.8) to the inlet flow. The vessel Alls, the flow rate increases, but the residence time distribution is constant. 

Washout experiments can be used to measure the residence time distribution in continuous-flow systems. A good step change must be made at the reactor inlet. The concentration of tracer molecules leaving the system must be accurately measured at the outlet. If the tracer has a background concentration, it is subtracted from the experimental measurements. The flow properties of the tracer molecules must be similar to those of the reactant molecules. It is usually possible to meet these requirements in practice. The major theoretical requirement is that the inlet and outlet streams have unidirectional flows so that molecules that once enter the system stay in until they exit, never to return. Systems with unidirectional inlet and outlet streams are closed in the sense of the axial dispersion model i.e., Di = D ut = 0- See Sections 9.3.1 and 15.2.2. Most systems of chemical engineering importance are closed to a reasonable approximation. 

Example 15.3 Determine the first three moments about the origin and about the mean for the residence time distribution of a CSTR. 

This function is shown in Figure 15.9. It has a sharp first appearance time at tflrst = tj2. and a slowly decreasing tail. When t > 4.3f, the washout function for parabohc flow decreases more slowly than that for an exponential distribution. Long residence times are associated with material near the tube wall rjR = 0.94 for t = 4.3t. This material is relatively stagnant and causes a very broad distribution of residence times. In fact, the second moment and thus the variance of the residence time distribution would be infinite in the complete absence of diffusion. 

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