Tracer equations

When MN(C) is included as a variable the coefficient for PB became statistically insignificant (p = 0.36). The coefficient for PB in equation (15) was a factor of two smaller than in equation (16) however, this is not a significant difference in view of the uncertainties of the coefficients, particularly in equation (16). If additional variables (CU or MN ) were included in the multiple regression analysis, these were placed in the equations ahead of PB the coefficient for PB was not statistically significant in any of these equations and often negative. This can be seen in equation (16). If MN was used as a mixed auto/soil tracer, equation (17) was obtained. Although this does not allow separate estimates of the contributions of automobiles and soil resuspension, the particle-size of the tracer is more appropriate. 

Since the tracer equations are linear differential equations, a Laplace transform L f(t) = jf(t)e s,dt may be used to relate tracer inputs to responses. The concept of a transfer function facilitates the combination of linear elements. 

Surface tracer fluxes, such as freshwater fluxes, may be included by appropriate surface boundary conditions (Griffies et al., 2001). The tracer equations represent also the interface between the chemical-biological model components and the physical model. In this case, terms have to be added that describe sources and sinks due to chemical-biological reactions. 

In the case of a tritiated tracer, where there is homology between the structure of the ligand and the tracer, equations are identical and Ka = Ka... 

When the tracer aliquot contains only a weightless quantity of X, one arrives at the classical tracer equation... 

Of course, the typical objective is the Ki a value for a different gas, such as oxygen, to be used in such things as total daily maximum load (TMDL) calculations, and we just have the Ki a value for the gas tracer. Equation 9.22 can be used to get us from the transfer of one compound to another, because r is similar for all compounds ... 

Finally, it should be pointed out that all the results on dynamical testing presented here are correct only if the tracer gas is not significantly adsorbed at the solid surface. If it is adsorbed weakly, so that a linear isotherm is appropriate, the equations should be modified by the following replacements ... 

Equations 13.31 and 13.32 are only valid if the radioactive element in the tracer has a half-life that is considerably longer than the time needed to conduct the analysis. If this is not the case, then the decrease in activity is due both to the effect of dilution and the natural decrease in the isotope s activity. Some common radioactive isotopes for use in isotope dilution are listed in Table 13.1. 

If the source fingerprints, for each of n sources are known and the number of sources is less than or equal to the number of measured species (n < m), an estimate for the solution to the system of equations (3) can be obtained. If m > n, then the set of equations is overdetermined, and least-squares or linear programming techniques are used to solve for L. This is the basis of the chemical mass balance (CMB) method (20,21). If each source emits a particular species unique to it, then a very simple tracer technique can be used (5). Examples of commonly used tracers are lead and bromine from mobile sources, nickel from fuel oil, and sodium from sea salt. The condition that each source have a unique tracer species is not often met in practice. 

Caldwell-Babb Darken observed that sohd-state diffusion in metallurgical applications followed a simple relation. His equation related the tracer diffusivities and mole fractious to the mutual diffusivity ... 

Grinding-Rate Functions These were determined by tracer experiments in laboratoiy mills by Kelsall et al. (op. cit.) as shown in Fig. 20-19 and in similar work by Szantho and Fuhrmann [Aujhereit. Tech., 9(5), 222 (1968)]. These cuives can be fitted by the following equation ... 

For operation with an inert tracer, the material balances are conveniently handled as Laplace transforms. For a stirred tank, the differential equation... 

The unsteady material balances of tracer tests are represented by linear differential equations with constant coefficients that relate an input function Cj t) to a response function of the form... 

Kinds oi Inputs Since a tracer material balance is represented by a linear differential equation, the response to anv one kind of input is derivable from some other known input, either analytically or numerically. Although in practice some arbitrary variation of input concentration with time may be employed, five mathematically simple input signals supply most needs. Impulse and step are defined in the Glossaiy (Table 23-3). Square pulse is changed at time a, kept constant for an interval, then reduced to the original value. Ramp is changed at a constant rate for a period of interest. A sinusoid is a signal that varies sinusoidally with time. Sinusoidal concentrations are not easy to achieve, but such variations of flow rate and temperature are treated in the vast literature of automatic control and may have potential in tracer studies. 

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. 

These are the most basic equations needed in order to utilize U-Th series radionuclides as time-dependent tracers. In turn, they can provide an apparent age of a process or a geochronological framework. 

An approximate, but sufficient, solution of dris equation for the thin tracer him experimental technique shows that the penetration curve has two components Ai, All and b being known constants. 

Using Equation 8-51 it is possible to determine E(t) from die measured response C(t), if bodi die volumetric flowrate u and die mass of tracer injected ni are known. Since die mass of tracer emerging in die interval from t and t -i- 6t is C(t)u6t, dien die total quantity of die tracer is ... 

It can be seen that the equality defined in Equation 4.32 is true only when ICSo = KB (i.e., the concentration of a non-competitive antagonist that reduces the binding of a tracer ligand by 50% is equal to the equilibrium dissociation constant of the antagonist-receptor complex). 

In fluid reservoirs like fhe afmosphere or the ocean, the turnover time of a tracer is also related to the spatial and temporal variability of ifs concentration within the reservoir a long turnover time corresponds to a small variability and vice versa (Junge, 1974 Hamrud, 1983). Figure 4-2 shows a plot of measured trace gas variability in the atmosphere versus turnover time estimated by applying budget considerations as indicated by Equation (1). An inverse relation is obvious, but the scatter in the data... 

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