Mass balance estimation technique

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. 

A second major estimating technique is the materials balance approach—the original focus of this paper. A chemical engineering standard, the materials balance can reduce to the simple mass balance, as when the measured mass of a chemical in products leaving the plant is subtracted from the raw material entering the plant to yield the loss. This loss is then partitioned among releases to various media or other sinks. If... 

Various munerical techniques are used to indirectly obtain solutions to large systems of equations with too many imknowns to solve explicitly. One approach is to solve the equations iteratively. This is done by first assuming that all of the anions are unbound and, hence, their free ion concentrations are equal to their total (stoichiometric) concentrations. By substituting these assumed anion concentrations into the cation mass balance equations, an initial estimate is obtained for the free cation concentrations. These cation concentrations are substituted into the anion mass balance equations to obtain a first estimate of the free anion concentrations. These free anion concentrations are then used to recompute the free cation concentrations. The recalculations are continued imtil the resulting free ion concentrations exhibit little change with further iterations. The computer programs used to perform speciation calculations perform these iterations in a matter of seconds. 

PLS (partial least squares) multiple regression technique is used to estimate contributions of various polluting sources in ambient aerosol composition. The characteristics and performance of the PLS method are compared to those of chemical mass balance regression model (CMB) and target transformation factor analysis model (TTFA). Results on the Quail Roost Data, a synthetic data set generated as a basis to compare various receptor models, is reported. PLS proves to be especially useful when the elemental compositions of both the polluting sources and the aerosol samples are measured with noise and there is a high correlation in both blocks. 

Now if we have data of C/Cj. as a function of pipe radius, r, we can use standard least-squares techniques to estimate Ko, Ki, K, . In addition, we can find the standard deviations of the estimates of Ki by the least-squares procedure, which gives an indication of the precision of the data. The first constant, Ko, should be unity if we have a perfect mass balance, and the deviation from this value gives an estimate of the reliability of the data. Knowing the injection tube size, we can find the Ni/q from the least squares K from Eq. (59). 

The isotope dilution results in Table II are on fuel source samples obtained from NBS which were considered homogeneous. The results in Table III are from the sampling points indicated in Figure 4. These summarized results are mostly by the SSMS general scan technique which has an estimated accuracy of better than 50%. The isotope dilution measurements are limited by the emulsion detector to 3-5%. The results are in grams of metal flow per minute. The mass balance for the various elements was computed by the following equations ... 

Ionization of the oxide/water interface and the resultant electrical double layer have been studied intensively by a variety of techniques within the last decade. Although many electrical double layer and adsorption models have been proposed, few are sufficiently general to consider surface equilibria in complex electrolyte solutions. Recently we proposed a comprehensive adsorption model for the oxide/water interface which can simultaneously estimate adsorption density, surface charge, and electro-kinetic potential in a self-consistent manner (jL, 2, 3). One advantage of the model was that it could be incorporated within the computer program, MINEQL ( ), by adding charge and mass-balance equations for the surface. 

Quantitative SPECT regimes are well established and rely on fewer assumptions about, e.g., three-dimensional activity distribution than planar activity quantitation approaches. While phantom studies can be used to assess accuracy of activity quantitation, these may not entirely reflect the accuracy achieved in human subjects. Mass balance calculations, where the initial activity loaded into the delivery device is compared to the activity in the subject estimated by the imaging technique plus the residual activities in the device and exhalation filter, can provide a good... 

Application ofGIS techniques for calculation and mapping of critical loads Critical loads ofpollutants at an ecosystem can be calculated on the basis of the Steady-State Mass Balance (SSMB) biogeochemical model (see Chapter 10). All equations of this model include a quantitative estimation of the greatest possible number of... 

Once a PBPK model is developed and implemented, it should be tested for mass balance consistency, as weU as through simulated test cases that can highlight potential errors. These test cases often include software boundary conditions, such as zero dose and high initial tissue concentrations. Some parameters in the PBPK model may have to be estimated through available in vivo data via standard techniques such as nonlinear regression or maximum likelihood estimation (30). Furthermore, in vivo data can be used to update existing (or prior) PBPK model parameter estimates in a Bayesian framework, and thus help in the rehnement of the PBPK model. The Markov chain Monte Carlo (MCMC) (31-34) is one of the... 

A TRIAX 550 spectrometer attached to an Andor -90°C cooled CCD detector was used for all spectroscopic measurements. Ar laser lines at 488.0 nm and 514.3 nm were used. Reactions were monitored by time-resolved Raman spectroscopy, sequentially setup for two of three separate regions of interest within the spectral range. This enabled, for example, collecting information about the carboxylation/decarboxylation and hydration/dehydration processes by monitoring the various CO and CH vibration modes. This technique provided spectra in each region only after the collection of the spectra in other regions, and hence not favorable for faster kinetics. However, inclusion of OH and H2 peaks gave a reasonably quantitative estimate on the extent of the hydrothermal reaction and valuable information for mass balance calculations (see further details in the experimental results for each system)... 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

Finally, Rushing et al. have created a mass balance approach that directly integrates dynamic elements of the total hydrocarbon system with all static reservoir engineering, geochemical and petrophysical components. This prospect evaluation technique provides an independent estimate of hydrocarbon resource-in-place for comparison with that computed by the reservoir engineering approach. 

To address these problems, this paper presents a mass balance technique specifically developed for evaluating the resource-in-place potential of basin-centred gas prospects. This paper begins with a general overview of basin-centred gas systems (BCGS), including a summary of common attributes identified from a literature survey. A derivation of the mass balance technique and an explanation of its elements foUow this summary. Application of the technique is illustrated with an example from the Bossier tight gas sand play in the East Texas Basin located in eastern Texas, USA. Uncertainty in resource-in-place estimates are quantified by incorporating a Monte Carlo simulation technique with the mass balance computations. 

Note that this mass is computed independently from and without consideration of the reservoir rock and fluid properties. Consequently, this hydrocarbon mass should provide an excellent validation of that computed from the eonven-tional reservoir engineering approach. Moreover, the difference between the hydrocarbon mass remaining in the reservoir and the mass balance and conventional reservoir engineering techniques provides an estimate of the volume of generated hydrocarbon that has escaped from the reservoir because of insufficient seals or traps. 

Specifically, this technique was used to estimate the range of gas-in-place volumes—i.e. the key parameter—and their probabihty of oecurrence. Equations (12) and (14) are the models defining the gas volumes computed from the mass balance and conventional reservoir engineering techniques, respectively. 

A classic technique employed in pharmacology and toxicology disposition studies for all routes of administration is the mass balance approach (Riviere, 1999). Mass balance analysis accounts for all of the topically applied dose of the compound, whether it is in the formulation, associated with the skin surface, penetrated into the stratum comeum, distributed into the carcass, or absorbed into and excreted from the blood into urine and feces. In this context, total recovery of 90% of the apphed dose is considered excellent recovery (Schaefer and Redelmeier, 1996). Mass balance studies are conducted by collecting all excreta after topical and parenteral administration. Data from a parenteral route such as intravenous dosing is required to correct for the fraction of absorbed compoimd appearing into the excreta collected if a precise estimate of bioavailability is to be determined and all routes of excretion are not collected (e.g., collection of urine and feces but not expired air) (Riviere, 1999). In such a study, absorption is calculated as follows ... 

---