Residence-time distribution

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

A theoretically derived equation for laminar flow in helical pipe coils by Ruthven (Chem. Eng. Sci, 26, 1113-1121 [1971] 33, 628-629 [1978]) 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 DL = l.OlvRe0 875 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. 

Noncircular Channels Calculation of frictional pressure drop in noncircular channels depends on whether the flow is laminar or turbulent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter DH should be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraulic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraulic diameter for a circular pipe is DH = D, for an annulus of inner diameter d and outer diameter D,DH = D-d, for a rectangular duct of sides a, b, DH=ab/[2(a+b)]. The hydraulic radius Rh is defined as one-fourth of the hydraulic diameter. 

With the hydraulic diameter subsititued for D in f and Re, Eqs. (6-37) through (6-40) are good approximations. Note that V appearing in / and Re is the actual average velocity V = Q/A for noncircular pipes it is not Q/(7cDh/4). The pressure drop should be calculated from the friction factor for noncircular pipes. Equations relating Q to AP and D for circular pipes may not he used for noncircular pipes with D replaced by DH because V Q/(nDfi/4). 

Appropriate equipment design is one way to guarantee the desired product properties by manipulation of the residence time of particles in fluidized bed processes. Moreover, it is necessary to identify suitable process inputs with an influence on residence time for use in feedback control loops. 

In order to quantify the influence of the gas flow rate on the residence time of the particles a simple model can be used that represents the horizontal apparatus by a series of continuously operated stirred tank reactors (CSTRs). The principle of this model is illustrated in Fig. 7.40. The size (length) and number of the tanks express the intensity of back-mixing (mixing in the direction of solids transport). They are flctitious for an open process chamber (as in Fig. 7.38), but may correspond to 

The average residence time under continuous, steady-state conditions results from the mass of particles in the fluidized bed Mbed and the mass flow rate of solids Mg.  

Using a correspondingly defined dimensionless residence time 

To find out the effective number of compartments that corresponds to the horizontal equipment of Fig. 7.38, tracer experiments were conducted. After a prescribed time at steady operating conditions a granular tracer substance was added to the solid feed stream. Samples were taken at the discharge of the plant at defined time intervals and analyzed by means of pH measurement after dissolution in de-mineralized water. The pH value of the solution corresponds to the concentration of tracer, and can be used to calculate residence time curves. Measurements were carried out for the two already mentioned total gas flow rates of 500 and 700 m h under otherwise the same conditions (mass flow rate of solids 20 kg h , bed mass 25 kg, tracer mass 5 kg, test duration 140-150 min). 

The RTD was first introduced by Danckwerts (1953), and it is deflned so that f(t)dt measures the fraction of the exit stream with residence time between t and t + dt. The cumulative RTD function or the F function, F t), is then defined as 

Example 6.8. Mean Residence Time and F(t) Function for Capillary Flow 

Calculate the mean residence time and the cumulative RTD function for pressure (Poiseuille) flow of a power-law fluid through a circular pipe (CPPF). 

Solution. The velocity profile is given in Table 2.6. Then the volumetric flow rate, dQ, with residence time between t and t + dt is given by the product of the area between the circles with radii r and r+dr and the velocity as 

The ratio of the residence times of fluid elements at radial distances r and 0 is 

The CSTR with complete mixing and the PFR with no axial mixing are limiting behaviors that can be only approached in practice. Residence time distributions in real reactors can be found with tracer tests. 

In the most useful form the test consists of a momentary injection of a known amount of inert tracer at the inlet of the operating vessel and monitoring of its concentration at the outlet. The data are used most conveniently in reduced form, as E — C/C0 in terms of tr — t/t, where 

C0 = initial average concentration of tracer in the vessel, t = VJV = average residence time. 

The plotted data usually are somewhat skewed bell-shapes. Some 

The sketch identifies the nomenclature Mean residence time  

Temperature and time as a function of composition are shown for two values of UA/Vr for 3 particular case represented by 

The residence time considers the time that each fluid element or group of molecules remains in the reactor it also depends on the velocity of the molecules within the reactor, and therefore the flow in the reactor. The residence time can be equal to the space time if the velocity is uniform within a cross section of the reaction system, as is the case of an ideal PFR. However, this situation is not the same for tank-type reactor, because the velocity distribution is not uniform. In most nonideal reactors, residence time is not the same for all molecules. This result in variations in concentration along the reactor radial, i.e., its concentration inside and outlet tank reactors are not uniform. This means a need to define the residence time and calculate their distribution for each system. 

One can visualize RTD through an experiment with the use of chemical tracers, introducing them at a particular moment or since the beginning of the reaction. This chemical tracer should necessarily be a compound not reactive to the system under study, by measuring its concentration in the reactor outlet. In general, dye compounds are used, but also other materials can be used with conductive or radioactive material properties that can be measured quantitatively. 

From this, on integration between at t = 0, C = 0, and at t O, Gout = C, and considering r as space time  

The concentration of tracer at the outlet varies exponentially with time, indicating a variation of the distribution in the reactor. Thus, the molecules have different residence times. This is the distribution of concentration within an ideal tank reactor. 

Generally, the concentration is related to a function of RTD and assuming that a fraction of molecules having a residence time between the time interval t and t + At. At the instant t, the concentration of tracer at the outlet is C. Thus, we measured a fraction of molecules that remained in the reactor in a time less than t and another fraction that remained in the reactor for a time longer than t. The first fraction is represented by the cumulative distribution function F(t) and the second fraction is represented by the difference (1 — P(t)). This last fraction Co does not contain at the output of the reactor. 

Reactor performance in terms of selectivity and conversion or space-time yield depends on the hydrodynamics, mass and heat transfer of the process (see, e.g., the literature cited in [2]). In the following, the impact of the residence time distribution, mixing and heat transfer is discussed, either from the viewpoint of modeling or where available illustrated with examples. 

The influence of residence time distribution (RTD) on performance, selectivity and yield is the same in microreactors as in conventional reactors. Therefore, the eflfects are well understood. Nonetheless, the demonstration of this at the microscale has hardly been reported so far. However, some experimental techniques have been developed to measure RTDs in microchannel flows which allow comparison between different types of flows or flows run at different parameters so that at least optimal flow conditions with regard to RTD can be found. 

Laminar flows in microchannels with their parabolic velocity profiles face superposition by radial and axial diffusion, as described by Taylor and Aris [61], introducing a global axial dispersion coefficient Dax which is used as a parameter in their dispersion model [12]  

Following the dispersion model, RTD can be characterized by the Bodenstein number. Bo  

For typical lengths of microdiaimels in the range of a few centimeters and average residence time in the range of a few seconds, axial back-diffusion is negligible, so that RTD is governed by the ratio of radial diffusion time to the fluid dynamic residence time [12]  

The liquid RTD in monoliths was measured in the set-up described above (see Fig. 8.16). Monohths with a length of 50-100 cm were used for these experiments, and an additional length of 1 cm was investigated to take into account the effects of the tracer injection, hquid distribution, and collection section on residence time. 

The liquid (tap water) was pumped through the tracer injection fitting, where a determined tracer volume was injected. Before entering the spray nozzle, the liquid flow was monitored for the dye concentration using an optical flow cell. The liquid passed the channels of the monolith and was collected with a small stirred vessel (mixing-cup). From this mixing cup the liquid was led on to a second probe, which measured the dye concentration by spectroscopy. The absorption of light is related to the concentration of the tracer by the Beer-Lambert law. 

8 Reactive Stripping in Structured Catalytic Reactors Hydrodynamics and Reaction Performance 

Each experiment was repeated at least four times, and good reproducibility was found between the different experiments. Air was used as medium for the gas phase, but it transpired that gas flow has almost no influence on the liquid RTD. 

The RTD of katapak-S has been measured by Moritz and Hasse (Fig. 8.22). Below the load point, a narrow distribution with a pronounced tail is observed which can be attributed to stagnant zones present at low liquid loads. Around the load point, the continuous flow through the catalyst-filled bag causes a narrow distribution without any tailing, whereas the bypassing flows occurring beyond the load 

In the previous chapters we have been using simplified forms of these general equations to obtain the equations we have derived and solved previously. We now examine in a bit more detail the solutions to them in more realistic situations. 

We will not attempt to solve the preceding equations except in a few simple cases. Instead, we consider nonideal reactors using several simple models that have analytical solutions. For this it is convenient to consider the residence time distribution (RTD), or the probability of a molecule residing in the reactor for a time f. 

For any quantity that is a function of time we can describe its properties in terms of its distribution function and the moments of this function. We first define the probability distribution function p(t) as the probability that a molecule entering the reactor wih reside there for a time t. This function must be normalized 

The average value of the time t spent in the reactor is our definition of the residence time r, which is given by the expression 

All these statistical quantities are introduced in courses in statistics and analysis of data, but we will only need them to describe the effects of RTD, and will just use them iri these forms without further explanation. 

General spelll Possible spelling error new symbol name cblist is similar to existing symbol calist.  

After approximately 30CSTRs in series the result is the same as one PFR of equal volume. This makes sense mathematically in terms of our analysis and it also makes good sense intuitively because we are using the same total volume more efficiently. 

We first encountered in Chapter 3 on mixing in multicomponent systems the problem of bypassing and less than perfect mixing. If we have two or more reactants that must mix in order for reaction to occur, then any deviations from a single-valued residence time distribution will show up as an apparent deviation from the predictions based upon perfect mixing. The spread in the residence time distribution leads to different extents of reaction for the fluid elements with these different times. 

Chapter 9 Continuous Stirred Tank and the Piug Flow Reactors 

We have developed the equations for a steady-state CSTR in which the reversible reaction of A and B produces D and one mole of D reacts back to produce A and B but the kinetics are second order  

The geometric surface of the microchannels can be increased for performing catalytic reactions. Porous coatings are typically applied for this purpose. The porous layer can be catalytically active or serve as a support for a catalytic active phase. Different coating techniques are developed and tested over the last years as there is a steady increase of the number of catalytic appUcations in pharmaceutical and fine chemical industries driven by strict environmental regulations and policies introduced during the last decade [1]. 

This chapter presents an overview of the fundamentals of design and operation of single-phase and multiphase catalytic microreactors. Various designs are discussed including their advantages in specific catalytic processes. 

If the target product is an intermediate in sequential reactions, selectivity and yields are quite sensitive to the residence time. The axial dispersion in a tubular microreactor can often be described by a dispersion model. This model assumes that the RTD results from piston flow on which axial dispersion is superimposed. The latter is taken into account by means of an 

Multiphase Catalytic Reactors Theory, Design Manufacturing, and Applications, Fiist Edition. Edited by Zeynep Esen Onsan and Ahmet Kerim Avci. 2016 John Wiley Sons, Inc. Published 2016 by John Wiley Sons, Inc. 

The effective axial dispersion coefficient in laminar flow in cylindrical microchannels [6, 7] is given as follows  

The RTD can be determined from the velocity profiles in the channel. The axial flow in a screw extruder is a pressure flow because there are no axial velocity components of the screw or barrel. As a first approximation, the axial pressure flow can be considered a flow between parallel plates. The velocity profile for pressure flow of a power law fluid between parallel plates is (see Fig. 7.109)  

Coordinate y can be expressed as a function of time by the following substitution  

With this substitution, the external RTD function can be written as  

The cumulative RTD function F(t) can be found by integrating the external RTD function it can be expressed as  

If we express the RTD as a function of the dimensionless residence time 9, where 6 Is the actual residence time divided by the mean residence time, we get  

Direct and indirect approaches more commonly used for determination of filler compound quality are summarised in Table 5.1, together with pertinent measured parameters and influencing factors. Further discussion of characterisation procedures follows in Section 5.4.2. 

Screen pack analysis (pressure filter test) Filler dispersion 

Light/electron microscopy Filler dispersion/distribution 

Rheometry (capillary/dynamic/melt-flow index) Filler dispersion 

Infrared spectroscopy Chemical effects from polymer degradation 

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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... 

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. 

An industrial chemical reacdor is a complex device in which heat transfer, mass transfer, diffusion, and friction may occur along with chemical reaction, and it must be safe and controllable. In large vessels, questions of mixing of reactants, flow distribution, residence time distribution, and efficient utilization of the surface of porous catalysts also arise. A particular process can be dominated by one of these factors or by several of them for example, a reactor may on occasion be predominantly a heat exchanger or a mass-transfer device. A successful commercial unit is an economic balance of all these factors. 

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

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. 

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