Tracer distribution models

Although the general circulation patterns are fairly well known, it is difficult to quantify the rates of the various flows. Abyssal circulation is generally quite slow and variable on short time scales. The calculation of the rate of formation of abyssal water is also fraught with uncertainty. Probably the most promising means of assigning the time dimension to oceanic processes is through the study of the distribution of radioactive chemical tracers. Difficulties associated with the interpretation of radioactive tracer distributions lie both in the models used, nonconservative interactions, and the difference between the time scale of the physical transport phenomenon and the mean life of the tracer. 

Siegenthaler, U. and Joos, F. (1992). Use of a simple model for studying oceanic tracer distributions and the global carbon cycle, Tellus 44B, 186-207. 

Experimental residence time studies are to be carried out in which the solid in the bottom half of the bed is initially mixed with tracer, the bed started and timed samples taken from various locations in the bed annular region. It is desired to model the resulting tracer distribution in the bed in order to find the fraetional slippage rate between the annular tanks and the central bed regions. 

In the box-model the aqueous domain of the oceans is treated as a box in which the tracer distribution is assumed to be... 

Yet, Okubo s law and the physical model on which it is based disregard two important properties of measured tracer distributions. The first one concerns the shape of tracer clouds. As indicated in Fig. 22.9, clouds usually develop into elongated structures which can be approximated by ellipses with major and minor principal axes, cma and cmi. The major axis points in the direction of the mean flow. When Okubo... 

Maier-Reimer, E. (1993). Geochemical cycles in an ocean general circulation model - Preindustrial tracer distributions. Global Biogeochem. Cycles 7, 645—677. 

An important characteristic of a property distribution is encapsulated in the Peclet number, Pe = ULIk, which is the ratio of diffusive time-scale to advective timescale of the system. In this definition, U and L are the characteristic velocity and length scales of the flow. The Peclet number is a measure of the relative importance of advection versus diffusion, where a large number indicates an advectively dominated distribution, and a small number indicates a diffuse flow. Numerical modeling indicates that certain tracer distributions, in particular tracer-tracer relationships, are significantly affected by the Peclet number, and consequently can be used to determine the nature of the fluid flow (Jenkins, 1988 Musgrave, 1985, 1990). 

This new approach is currently receiving considerable attention in oceanography, and is a useful adjunct to the traditional strategies for analysis of tracer distributions and numerical models. A number of illustrative examples have been discussed in the literature (Deleersnijder ef al., 2001, 2002 Delhez et al, 2003 Haine and Hall, 2002 Hall and Haine, 2002). 

Diagnostic calculations, using observed tracer distributions (e.g., tracer or age gradients, or relationships with other tracers) it may be possible to calculate mixing, velocity or ventilation rates directly within the context of simple advective-diffusive or box models. 

Gudiksen et al. (1968) 0.15-3.6 Two-dimensional model including mean motions of 185W tracer distribution... 

Once calculated, the simulated tracer distributions can be compared with measurements. Traditionally, this model/data comparison is done subjectively by analyzing the misfit fields and hypothesizing possible causes for the misfits. Model flows or biogeochem-ical parameters are then modified, hoping that new simulations with the modified parameters lead to more realistic tracer fields. This manual tuning has been used successfully with small box models however, for problems with many thousands of parameters, such as the one described above, it is impractical and not successful in most cases. 

A scheme of the estimation of the spatial velocity distribution from the measured tracer distribution is given and the several concepts are presented for the utilization of these dates for the modelling of hydrodynamic flow processes in soil columns ... 

PET-Measurements of Tracer Distribution in the Model Soil Column... 

Real reactors deviate more or less from these ideal behaviors. Deviations may be detected with re.sidence time distributions (RTD) obtained with the aid of tracer tests. In other cases a mechanism may be postulated and its parameters checked against test data. The commonest models are combinations of CSTRs and PFRs in series and/or parallel. Thus, a stirred tank may be assumed completely mixed in the vicinity of the impeller and in plug flow near the outlet. 

The use of various statistical techniques has been discussed (46) for two situations. For standard air quality networks with an extensive period of record, analysis of residuals, visual inspection of scatter diagrams, and comparison of cumulative frequency distributions are quite useful techniques for assessing model performance. For tracer studies the spatial coverage is better, so that identification of meiximum measured concentrations during each test is more feasible. However, temporal coverage is more limited with a specific number of tests not continuous in time. 

Ross (R2) measured liquid-phase holdup and residence-time distribution by a tracer-pulse technique. Experiments were carried out for cocurrent flow in model columns of 2- and 4-in. diameter with air and water as fluid media, as well as in pilot-scale and industrial-scale reactors of 2-in. and 6.5-ft diameters used for the catalytic hydrogenation of petroleum fractions. The columns were packed with commercial cylindrical catalyst pellets of -in. diameter and length. The liquid holdup was from 40 to 50% of total bed volume for nominal liquid velocities from 8 to 200 ft/hr in the model reactors, from 26 to 32% of volume for nominal liquid velocities from 6 to 10.5 ft/hr in the pilot unit, and from 20 to 27 % for nominal liquid velocities from 27.9 to 68.6 ft/hr in the industrial unit. In that work, a few sets of results of residence-time distribution experiments are reported in graphical form, as tracer-response curves. 

Nuclear bomb produced " 002 and (as HTO) have been used to describe and model this rapid thermocline ventilation (Ostlund et ah, 1974 Sarmiento et ah, 1982 Fine et al., 1983). For example, changes in the distributions of tritium (Rooth and Ostlund, 1972) in the western Atlantic between 1972 (GEOSECS) and 1981 (TTO) are shown in Fig. 10-10 (Ostlund and Fine, 1979 Baes and Mulholland, 1985). In the 10 years following the atmospheric bomb tests of the early 1960s, a massive penetration of F1 (tritium) into the thermocline has occurred at all depths. Comparison of the GEOSECS and TTO data, which have a 9 year time difference, clearly shows the rapid ventilation of the North Atlantic and the value of such transient" tracers. A similar transient effect can be seen in the penetrative distribution of manmade chlorofluorocarbons, which have been released over a longer period (40 years) (Gammon et al., 1982). 

Fung, I. Y. (1986). Analysis of the seasonal and geographical patterns of atmospheric CO2 distributions with a three-dimensional tracer model. In "The Changing Carbon Cycle A Global Analysis" (J. R. Trabalka and D. E. Reichle, eds), pp. 459-473. Springer-Verlag, New York. 

As mentioned in Section 11.3, fluidized-bed reactors are difficult to scale. One approach is to build a cold-flow model of the process. This is a unit in which the solids are fluidized to simulate the proposed plant, but at ambient temperature and with plain air as the fluidizing gas. The objective is to determine the gas and solid flow patterns. Experiments using both adsorbed and nonadsorbed tracers can be used in this determination. The nonadsorbed tracer determines the gas-phase residence time using the methods of Chapter 15. The adsorbed tracer also measures time spent on the solid surface, from which the contact time distribution can be estimated. See Section 15.4.2. 

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. 

Transient experiments with inert tracers are used to determine residence time distributions. In real systems, they will be actual experiments. In theoretical studies, the experiments are mathematical and are applied to a d5mamic model of the system. 

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. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

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