Steady tracers

Steady tracers which are introduced continuously in a system, e.g., those produced by nuclear interactions of cosmic rays on the earth, and by radioactive decay of dissolved uranium in the oceans. 

Typically, there are two ways to inject tracers, steady tracer injection and unsteady tracer injection. It has been verified that both methods lead to the same results (Deckwer et al., 1974). For the steady injection method, a tracer is injected at the exit or some other convenient point, and the axial concentration profile is measured upward of the liquid bulk flow. The dispersion coefficients are then evaluated from this profile. With the unsteady injection method, a variable flow of tracer is injected, usually at the contactor inlet, and samples are normally taken at the exit. Electrolyte, dye, and heat are normally applied as the tracer for both methods, and each of them yields identical dispersion coefficients. Based on the assumptions that the velocities and holdups of individual phases are uniform in the radial and axial directions, and the axial and radial dispersion coefficients, E and E, are constant throughout the fluidized bed, the two-dimensional unsteady-state dispersion model is expressed by... 

Burgess et al." describe a study of gas storage cabinets. In the study, coefficient of entry (CJ for various inlet/outlet configurations was measured. A tracer gas study is also described. The tracer gas study involved releasing sulfur hexafluoride (SF ) at 0.032 L s" at a critical leak position in the cabinet and measuring SFg concentration in the exhaust stream. The tracer gas was turned off when a steady exhaust stream concentration was observed and the time for the concentration to decay to 5% of steady state was measured. 

In this example, an initial steady-state solution with a = 0 is propagated downstream. At the fourth axial position, the concentration in one cell is increased to 16. This can represent round-off error, a numerical blunder, or the injection of a tracer. Whatever the cause, the magnitude of the upset decreases at downstream points and gradually spreads out due to diffusion in the y-direction. The total quantity of injected material (16 in this case) remains constant. This is how a real system is expected to behave. The solution technique conserves mass and is stable. 

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

The Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

The pulse tracer response, giving the E-curve response, can be used directly to calculate the steady-state conversion for a first-order reaction according to the relationship... 

This program is designed to simulate tracer experiments for residence time distributions based on a cascade of 1 to 8 tanks-in-series. An nth-order reaction can be run, and the steady-state conversion can be obtained. The important parameters to change are as follows for the tracer experiments k, CAINIT, and CAO ( = 0 for E curve, = 1 for F curve). For reaction studies, the parameters to change are n, k, CAO, and CAINIT. 

Nozaki Y, Yamada M, Nikaido H (1990) The marine geochemistry of actinium-227 evidence for its migration through sediment pore water. Geophys Res Lett 17 1933-1936 Nozaki Y (1993) Actinium-227 a steady state tracer for the deep-se basin wide circulation and mixing studes. In Deep Ocean Circulation, Physical and Chemical Aspects. Teramoto T (ed) Elsevier p 139-155... 

In Section 11.1.3.1 we considered the longitudinal dispersion model for flow in tubular reactors and indicated how one may employ tracer measurements to determine the magnitude of the dispersion parameter used in the model. In this section we will consider the problem of determining the conversion that will be attained when the model reactor operates at steady state. We will proceed by writing a material balance on a reactant species A using a tubular reactor. The mass balance over a reactor element of length AZ becomes ... 

The bed was first operated at the preselected conditions at a steady state then about 455 kg of the coarse crushed-acrylic particles, similar to that used as the bed material but of sizes larger than 6-mesh, were injected into the bed as fast as possible to serve as the tracer particles. Solids samples were then continuously collected from five different sampling locations at 30-second intervals for the first 18 minutes and at 60-second intervals thereafter. The samples were then sieved and analyzed for coarse tracer particle concentration. Typical tracer particle concentration profiles vs. time at each sampling location are presented in Figs. 38-42 for set point 3. 

Typically it took about 160 to 200 seconds to inject a pulse of about 455 kg coarse tracer particles into the bed pneumatically from the coaxial solid feed tube. It can be clearly seen from Figs. 38 to 42 that the tracer particle concentration increases from essentially zero to a final equilibrium value, depending on the location of the sampling port. The steady state was usually reached within about 5 minutes. There is considerable scatter in the data in some cases. This is to be expected because the tracer concentration to be detected is small, on the order of 4%, and absolute uniformity of mixing inside a heterogeneous fluidized bed is difficult to obtain. 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

To further test the model, calculations were performed to simulate the isotopic tracer experiments presented in Figs. 9 and 10. It should be noted that while the tracer experiments were performed at 438K, the rate coefficients used in the model were chosen to fit the experiments in which chemisorbed NO was reduced at 423 K. Figures 21 and 22 illustrate the nitrogen partial pressure and surface coverage responses predicted for an experiment in which 5 0 is substituted for l NO at the same time that H2 is added to the NO flow. Similar plots are shown in Figs. 23 and 24 for an experiment in which NO is substituted for during steady-state reduction. 

Consider a small volume of fluid q a entering the vessel virtually instantaneously over the time interval dt at a particular time (t = 0). Thus q 0 = qdt, such that q V and dt t. We note that only the small amount q 0 enters at t = 0. This means that at any subsequent time t, in the exit stream, only fluid that originates from q is of age f to t + dt all other elements of fluid leaving the vessel in this interval are either older or younger than this. In an actual experiment to measure E(t), q g could be a small pulse of tracer material, distinguishable in some manner from the main fluid. In any case, for convenience, we refer to q 0 as tracer, and to obtain E(t), we keep track of tracer by a material balance as it leaves the vessel. Note that the process is unsteady-state with respect to q 0 (which enters only once), even though the flow at rate q (which is maintained) is in steady state. 

To develop E(B) for two CSTRs in series, we use a slightly different, but equivalent, method from that used for a single CSTR in Section 13.4.1.1. Thus, consider a small amount (moles) of tracer M, nMo = F,dt, where Ft is the total steady-state molar flow rate, added to the first vessel at time 0. The initial concentration of M is cMo = nMo/(V/2). We develop a material balance for M around each tank to determine the time-dependent outlet concentration of M from the second vessel, cM2(l). 

A step input of tracer may itself be one of two types a step increase fiom one steady-state value to another (cAin), or a step decrease. Usually, as illustrated in Figure 19.4, tiie step increase is fiom a zoo value, and, as illustrated in Figure 19.5, the step decrease is to a zero value in the latter case it is called a washout. Figures 19.4 and. 5 also show responses (cA out) to a step increase and to a washout-step decrease. In Figure 19.4, note that cAout cA in as 1 °°> similarly in Figure 19.5, cAout — 0. 

A tracer study mas7 use a step increase followed at a later time by a step decrease the transient responses in the two cases are then checked far consistency. When considered separately, the washout technique has advantages less tracer is required, and it avoids having to maintain a steady-state value of cA in for a lengthy period. 

As discussed in Section 19.3.2.1, a step change (increase or decrease) in inlet concentration of a tracer to a vessel causes the outlet concentration to change over time from an initial steady-state level (say, cA1) to a second steady-state level (c ). As shown in Figure 19.4(a), cA1 may be zero for a step increase, or, as shown in Figure 19.5(a), cA2 is... 

For a step change, a material-balance criterion, analogous to equation 19.3-2 for a pulse input, is that the steady-state inlet and outlet tracer concentrations must be equal, both before and after the step change. Then, it may be concluded that the response of the system is linear with respect to the tracer, and that there is no loss of tracer because of reaction or adsorption. 

Consider the steady flow of a constant-density fluid at qg m3 s 1 through the N stirred tanks (Figure 19.11). At t = 0, a quantity of na moles of a nonreacting tracer A is introduced into the first tank as a pulse or Dirac input, S(t — 0) s S(t). At any subsequent time t. a material balance for tracer around the i th tank is ... 

Figure 19.15. We assume steady flow overall, but not with respect to (tracer) A there is no reaction taking place. In words, we write this as ...

The derivation of the material-balance or continuity equation for reactant A is similar to that of equations 19.4-48 and -49 for nonreacting tracer A, except that steady state replaces unsteady state (cA at a point is not a function of t), and a reaction term must be added. Thus, using the control volume in Figure 19.15, we obtain the equivalent of equation 19.4-48 as ... 

The boundary conditions for a closed-vessel reactor are analogous to those for a tracer in a closed vessel without reaction, equations 19.4-66 and -67, except that we are assuming steady-state operation here. These are called the Danckwerts boundary conditions (Danckwerts, 1953).1 With reference to Figure 19.18,... 

The residence time characteristics of a reaction vessel are investigated under steady-flow conditions. A pulse of tracer is injected upstream and samples are taken at both the inlet and outlet of the vessel with the following results ... 

Tracer input to a CSTR is sinusoidal with equation C/Cf - l+sin((Jt). Transient and steady output concentrations are to be found. The unsteady material balance is... 

A system of N continuous stirred-tank reactors is used to carry out a first-order isothermal reaction. A simulated pulse tracer experiment can be made on the reactor system, and the results can be used to evaluate the steady state conversion from the residence time distribution function (E-curve). A comparison can be made between reactor performance and that calculated from the simulated tracer data. 

Information from tracer experiments can be used to calculate the steady state conversion according to the following relationship (Levenspiel, 1999)... 

A comparison of the effects of axial and radial mixing is seen in Figure 4.7, which shows results obtained by Gunn and Pryce(28) for dispersion of argon into air. The values of Dl were obtained as indicated earlier, and DR was determined by injecting a steady stream of tracer at the axis and measuring the radial concentration gradient across the... 

Equation (9.27) defines the so-called axial dispersion coefficient Dax as a model parameter of mixing. Nd is the dispersion flow rate, c the concentration of the tracer mentioned earlier, and S the cross-sectional area of the column. The complete mole flow rate of the tracer consists of an axial convection flow and the axial dispersion flow. The balance of the tracer amount at a cross section of the extractor leads to second-order partial differential equations for both phase flows at steady state. For example, for continuous liquids ... 

Imagine that a reactor is operating in the steady state with a constant throughflow, Q, of 3.7 dm min. At the reactor inlet, 50 g of concentrated dye are instantaneously injected. The outlet concentration of this tracer is detected and recorded as a function of time. Steady-state calibrations of the dye-detecting system show that its sensitivity is essentially constant and equal to 3.88 mV (g dye dm over the range of dye concentrations encountered. The data in Table 1 is collected. 

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