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There are two main ideal reactor models that are used for description of flow behaviour. The two categories of reactor are the continuous stirred tank reactor (CSTR) and the plug flow reactor (PFR). In an ideal CSTR the reactant concentration is uniform throughout the vessel. The formation of dead zones, causing reactant concentration to be low, or feed points, causing reactant concentration to be relatively high, result in non-ideal conditions. In an ideal PFR all reactant and product molecules have the same flow rate at any position along the axis. Deviations arise due to turbulent mixing or molecular diffusion which causes molecules to move with changing speeds and in different directions. The deviations from ideal reactor conditions pose several problems in the design and analysis of reactors. Figure 4. presents the possible deviations from ideality namely short circuiting or by pass reactant flows into the tank through to the outlet without reacting if the inlet and outlet are close by or if there exists an easy route between the two. To describe these deviations from ideality, three concepts are generally used, which are regarded as characteristic of mixing (Levenspiel, 1999):
The residence time distribution (RTD).
The quality of mixing.
The model used to describe the process.
Figure 4.: Real flow patterns exist in process equipment (Levenspiel, 1999).
The residence time is the time an element of fluid spends inside the tank. Each element may spend a different time inside the tank and that distribution of elements goes to make up the residence time distribution. RTD can be used to characterise and diagnose the flow in a nonideal reactor. It is also considered a useful tool in the designing and modelling processes and can be obtained by the so-called "stimulus response" technique for a flowing fluid in a CSTR and PFR (Shah, 1979). The principle idea of this technique is to inject a tracer in form of a step or pulse at the entrance of the tank and measuring of its concentration at the outlet as a function of time (Levenspiel, 1999). A proper choice of this tracer can be outlined by the following conditions:
The physical properties of the tracer should be similar or closely follow those of the reacting mixture under treatment and be completely soluble in the mixture.
Not react with or be absorbed by the surface of the system.
Easily detectable in small concentrations.
The tracer should not change from phase to phase during the experiment.
The cost of tracer itself and its detection device should be relatively cheap.
The heat balance should be taken into account if heat is used as a tracer.
A radioactive tracer offers an advantage for measuring the RTD of a very fast-moving phase as a scintillation detection counter can be interfaced with very rapid recording systems.
In the liquid phase, usually the tracer (e.g. NaCl, H2SO4, etc.) can be detected by directly inserting a probe into the reactor and continuously monitoring the concentration by means of electrical conductivity. For the solid phase, a magnetic tracer may be used. The concentration of a solid-phase tracer can also be measured by a capacitance probe if the tracer has a different dielectric constant than the bulk solid phase. Radioactive tracers are convenient to use in the solid and gas phase, but radioactive tracer detection systems are expensive and must include scintillation counters, a recorder, etc. Table 4. presents some of tracers for liquid, gas and solid phases.
Table 4.:Tracer types employed in different phases and detection devices (Shah, 1979).
Liquid- phase tracers
Air- water glass beads
Gas- phase tracers
Air water glass Ballotini beads
Thermal conductivity cell
GowMac thermal conductivity detector
Glass spheres,air,ion-exchange resin, FeSiO2 powder, Cu powder
External sampling of solid-liquid soln. at various points
Radioactive tracer detector
Batch three-phase fluidized bed
Based on the data shown in Table 4. for an air-water system, sodium chloride has been selected as a tracer material to study the RTD for this thesis. Using this material, the detection device can measure the electrical conductivity of the effluent to determine the outlet concentration. Additionally, this tracer does not react or adsorb onto the adsorbent (NyexÂ®1000), is relatively cheap and is easily detectable in small amounts.
18.104.22.168 Measurement of RTD
A tracer technique is the experimental method for measuring RTD. This method has become a valuable technique in aiding the design of chemical reactors, aeration tanks for sewage treatment and in water pollution processes. Investigators of residence time distribution in continuous flow processes have applied various techniques to determine the tracer response in the outlet flow. Normally this technique involves the injection of known amounts of tracer solution in the form of a pulse, step, sinusoidal, or ramp in the inlet stream, followed by detecting its concentration as a function of time in the outlet stream. The position of injection and detection points should be very close to the reactor and no dispersion carries the tracer materials across system boundaries (i.e. only flow). The two most common methods used are pulse and step input, and the typical concentration curves at the inlet and outlet of the reactor are shown in Figure 4. (Fogler, 1999).
Figure 4.: RTD measurements technique (Fogler, 1999).
Pulse tracer input
A quantity of tracer is injected at the inlet of a system over a period of time, which is very short compared to the mean residence time, and its concentration is measured as a function of time in the outlet stream. Different conditions are assumed: constant flow rate and fluid density, only one phase flowing, no diffusion takes place across the system boundaries, and flat velocity profiles at the inlet and outlet. Furthermore, the magnitude of the tracer response at the outlet is directly proportional to the amount of tracer injected i.e. linearity with respect to the tracer analysis and the flow mixing behaviour of tracer is identical to the process fluid and is completely conserved within the system (Kayode Coker, 2001).
This technique has been investigated by many researchers. For example, Pulse input tracing has been performed by (Saravanathamizhan et al., 2008) to diagnose the electrolyte flow characteristics in a continuous stirred tank electrochemical reactor (CSTER) using RTD technique. Acid Red 88 dye solution was injected as a pulse input to the cell and the outlet concentrations have been analyzed as function to the time using a colorimeter. A three parameter model has been proposed to describe the electrolyte flow in a CSTER consisting of active, dead and bypass zones with exchange flow between dead and active zones.
The drawbacks of this technique are the injection must be done in very short time, the amount of tracer used should be known, and inaccuracy analysis could be take place when concentration versus time curve has a long tail.
Step tracer input
The step change residence time experiment can be performed if the feed to a system is switched from ordinary fluid to a fluid with tracer during the experiment and measuring its concentration versus time until the concentration of tracer at the exit of the system was the same as that of the feed tracer inlet. A step signal is often used if the tracer is cheap or pleasant (e.g. non radioactive tracers). Advantages of using a positive step in the RTD measurement are easier to carry out experimentally than a pulse test and the total amount of tracer injected in the stream feed over the period of the experiment does not have to be known as it does in the pulse test. On the other hand, the drawbacks to these techniques are difficult to maintain a constant tracer concentration in the feed, RTD function requires differentiation which can lead to error and large amount of tracer is required (Fogler, 1999).
22.214.171.124 Theoretical background of RTD
In pulse input moles of tracer is injected in one shot and the effluent concentration is measured. The amount of tracer material that has spent an amount of time between t and in the reactor is (Fogler, 1999),
where Q is the effluent volumetric flow rate
The fraction of material that has spent (residence time) in the reactor between t and for pulse input is:
where Et is the residence time distribution function of a real reactor also called the exit-age distribution function and equal to
If is not known directly, it can be calculated from outlet concentration measurements by summing up all the amount of material, so we can rewrite Equation (4.1) in integration form:
after integration Equation (4.3) and rearrangement, we obtain,
The integral in the denominator is the area under C-curve which is equal to and thus equal to, hence the exit age distribution become,
Another way to define the exit age distribution for the fraction of all the material that has spent a time from zero to infinity is equal 1; therefore,
So, the effective mean residence time (Equation 4.7) is compared to the expected value given by the ratio V/Q (V, is the total reactor volume). Specific problems of by-pass and/or dead (stagnant) volumes formed within the real reactor can be detected by these tests.
In step input the concentration of tracer is kept at this level till the outlet concentration equals the inlet concentration. The fraction of effluent which has been in the reactor for less than and is called the cumulative distribution function Ft.
Residence time distribution is probability density function, hence it can be characterise using statistical moments which are used to compare RTD's instead of comparing entire distribution (Wen and Fan, 1975). The mean residence time (), variance (), skewness (s3) and coefficient of variation (Cv) are statistical moments was defined as (Levenspiel, 1999); (Harris et al., 2003)
(if is constant) (4.9)
To describe probability distribution, coefficient of variation is normally used and defined as (Harris et al., 2003)
RTD in ideal reactors:
For CSTR, a material balance on tracer that has been injected in the form of a pulse input gives:
where is the space time and equal to V/Q and Co is initial concentration of tracer.
Substituting equation (4.13) in to equation (4.4) obtains:
Co is constant, hence Equation (4.14) becomes:
In order to directly compare the flow behaviour or performance inside reactors of different sizes, frequently a normalized RTD is used instead of the exit age distribution function, Et.
If the parameter Î¸ (equal to t/Ï„), which represents the number of reactor volumes of fluid based on entrance conditions that have flowed through the reactor in time.
Equation (4.9) has shown that generally the mean residence time in a reactor is equal to V/Q or Ï„. Applying this definition of a mean residence time to the RTD for a CSTR, gives:
In practical reactors the above equation (Equation 4.17) may not be valid and therefore will be a difference between the mean residence time and space time.
Applying the mean residence time to the second moment (variance) we obtain:
For PFR, all the molecules leaving a plug flow reactor have spent exactly the same amount of time within the reactor. The distribution function in PFR is a spike which has infinite height and zero width, whose area is equal to one. The spike can be represented mathematically by the Dirac delta function:
All the material has spent precisely the same time in the reactor, i.e. no variance, so
One parameter models:
Reactor models are useful for diagnosing flow behaviour and scale up in a real reactor. A range of model can be used to describe RTD. One parameter models that can be used to describe the mixing behaviour or interpret RTD deviations from two ideal reactors (CSTR, PFR) are tank in series model and dispersion plug flow model. For dispersion and tank in series models, Peclet number (Pe) and number of tank (nT) are the single parameters that can represent these deviation, respectively (Saravanathamizhan et al., 2010). These models preclude the requirement of applying computational fluid dynamics (CFD) by limiting their application to a single characteristic length (Martin, 2000).
The "dispersion model" is one of the most widely used models in RTD studies. Suppose an ideal pulse of tracer is injected into a plug flow reactor, the pulse spreads as it passes through this vessel. The tracer spreads upstream and downstream directions away from the centre of the original pulse due to molecular diffusion, a non-uniform velocity, and turbulent mixing. The longitudinal dispersion coefficient, D (m2 sï€1) represents this result in the tracer cloud. Thus, a large D indicates a rapid spreading of the tracer curve, a small D means slow spreading, and D = 0 means no spreading (hence plug flow). To characterize the spread in the whole reactor, the longitudinal dispersion number, D/uL, is usually used, where is the fluid velocity (m sï€1) and is the length of the reactor (m). This number has been evaluated from the shape of the tracer curve as it passes the exit of the vessel by measurement of the mean time () and the variance () from equations (4.9) and (4.10), respectively. Normally, the dispersion of axial mixing in streamline flow of fluid through pipes is mainly due to fluid velocity gradients, whereas radial mixing is due to molecular diffusion. To describe the dispersion in x-direction, Fick's diffusion law was used and defined as (Levenspiel, 1999),
In terms of the normalized form where and, the basic differential equation representing the dispersion model becomes
where 1 / Pe = is vessel dispersion number, and Pe is Peclet number which is equal to the rate of transport by convection (uL) divided to the rate of transport by or dispersion diffusion (D).
Two types of boundary conditions are considered, namely open and closed vessels. The first boundary condition is that in which the flow is undisturbed as it passes the inlet and outlet boundaries, whereas the secondary boundary condition is completely mixed and dispersion occurs between inlet and outlet boundaries.
Open dispersion model
The analytical solution for Equation (4.23) was published by (Levenspiel and Smith, 1957) for "open" boundary condition using dimensional group as shown in Equation (4.24).
With mean and variance
The subscript (o) indicates open boundary conditions.
Closed dispersion model
This system was treated with closed boundary conditions in which the flow approaches the inlet to the reactor in an idealised plug flow (), transforms to dispersed flow within the reactor and returns to idealised plug flow at outlet stream. The analytical solution for Equation (4.23) did not report by (Levenspiel, 1999) for closed system in terms of exit age distribution. However, the analytical solution for this equation was published by (Thomas and McKee, 1944) with closed boundary condition in a dimensionless form. Their solution was reproduced by (Yagi and Miyauchi, 1953) with an another summarizing of the terms.
where Î±n is the positive root of Equation (4.28).
The mean and variance of Levenspiel's results are shown in Equations 4.29 and 4.30, respectively.
where is given by Equation (4.31).
The subscript (c) denotes closed boundary conditions.
Tank in series model
This model describes the flow in CSTR by considering it to be discretised into a series of equal sized CSTRs; each of them is independent of those preceding or following it (Martin, 2000). Integration of a simple dynamic tracer mass balance on the nT stages of CSTR gives the system RTD as defined by
with mean and variance
The subscript (T) indicates the tank in series model.
In summary, the tank in series model is easier to apply compared to open dispersion model due to its relatively simpler mathematical definition and hence is applicable to any kinetics and any configuration of compartments with or without recycle (Martin, 2000). Additionally, defining the entry and exit boundary conditions is not critical as in the dispersion model. Practically, the suitability of this model improves as the number of tanks in series reduces thereby making the choice of boundary conditions (whether open or closed) important. However, this model has a significant drawback when nT is small due to the integer constraint, i.e. only whole numbers of tanks are allowed.