Uncertainties in calculations

Relevant hydrological fundamentals are utilized (21) to take account of the complex interaction of physical and chemical processes involving sod or rock, water, and contaminant. Attention is paid to uncertainties in calculated results. 

It is these contrasting effects of aerosol particles, combined with uncertainties in the contribution of absorption due to 03, that provide the largest uncertainties in calculations of actinic fluxes and photolysis rates in the boundary layer (e.g., Schwander et al., 1997). As a result, it is important to use the appropriate input... 

In summary, it may be concluded that the uncertainty in calculating absolute cross-sectional areas, the variation in cross-sectional areas with the BET C value and the fact that on porous surfaces less area is available for larger adsorbate molecules all point to the need for a universal, although possibly arbitrary, standard adsorbate. The unique properties of nitrogen have led to its acceptance in this role with an assigned cross-sectional area of 16.2 usually at its boiling point of —195.6 °C. 

Chapter 3 gave rules for propagation of uncertainty in calculations. For example, if we were dividing a mass by a volume to find density, the uncertainty in density is derived from the uncertainties in mass and volume. The most common estimates of uncertainty are the standard deviation and the confidence interval. 

The RMS error of Eq. 1.13 is related to the 2-norm of the vector of differences Pi — P°, while that of Eq. 1.14 is related to the oo-norm [1], We are not asserting that the latter should replace the former as an estimate of uncertainties in calculations, but that both error estimates should be considered if the uncertainties in predicted values for which there are no standards are to be realistic. We should note that a more elaborate approach to comparing computed results to a set of standard values has been devised by Maroulis and co-workers using principles of information theory [2]. Their work has been little used in practice, but it provides a convenient way of comparing many values. 

Only in situations where numerical uncertainty is small compared to modeling uncertainty we can successfully validate a calculation. After minimizing numerical errors there will still be other uncertainties in calculations due to for example variations in inlet conditions or due to inherent uncertainty in tabulated material properties, etc. These can be best handled by repeating the calculations with appropriate variations in the uncertain input quantities, thus resulting in say nc calculations with seemingly n, random outcomes the mean and variance of which are donated by Xc and. S 2, Similarly there would be ne repeated experiments of the same phenomenon with a, random outcomes with the corresponding mean and variance, Xe and S2e, respectively. The estimated modeling error is by definition the difference between the experimental mean and calculation mean, i.e. 

Error analysis. The mathematical analysis done to show quantitatively how uncertainties in data produce uncertainty in calculated results, and to find the sizes of the uncertainty in the results. [In mathematics the word analysis is synonymous with calculus, or a method for mathematical... 

Experimental errors tend to be quite large in biological systems, e.g. 30% in protein concentration measurements. Cell number measurements are generally no better than 5%. At lower viabilities (< 70% viable), accurate determination of viability and cell number is difficult, and the error in each determination may be greater than 10%. Errors in cell and metabolite concentration measurements lead to uncertainties in calculated parameters, such as specific growth, production and consumption rates, therefore a complex profile for these calculated parameters should not be assumed when a straight-line or simple function will suffice. 

The uncertainties in calculated flux values are large due to methodology errors and errors associated with extrapolating the collected data in space and time. A major problem with saltmarsh/ coastal water flux studies is that the flux is often smaller than the error margin, hence one is not even sure of the direction of the flux. The error associated with an annual flux estimate is much higher than the tidal flux. 

Regression analysis is often employed to fit experimental data to a mathematical model. The purpose may be to determine physical properties or constants (e.g., rate constants, transport coefficients), to discriminate between proposed models, to interpolate or extrapolate data, etc. The model should provide estimates of the uncertainty in calculations from the resulting model and, if possible, make use of available error in the data. An initial model (or models) may be empirical, but with advanced knowledge of reactors, distillation columns, other separation devices, heat exchangers, etc., more sophisticated and fundamental models can be employed. As a starting point, a linear equation with a single independent variable may be initially chosen. Of importance, is the mathematical model linear In general, a function,/, of a set of adjustable parameters, 3y, is linear if a derivative of that function with respect to any adjustable parameter is not itself a function of any other adjustable parameter, that is. 

Anderson, G.M., 1977. Uncertainties in calculations involving thermodynamic data, in H.J. Greenwood, ed. Applications of Thermodynamic to Petrology and Ore Deposits, Vancouver, Mineralogical Association of Canada, pp. 199-215. 

Fugacities were calculated from the Redlich-Kwong equation of state and there were differences of 30% between experiment and prediction, which reflects the uncertainties in calculating fugacities. 

The uncertainties in calculating specific wear rates from in-service data are discussed, as well as the data from laboratory tests which attempt to simulate particular modes of wear (eg abrasion, fretting). 

Figure 3.5 Dependence of uncertainty in calculated rate constant on the percentage change in the concentration of the species for which one analyzes and the precision of the chemical analysis. (Figure courtesy of Professor F. Tiscareno of the Institute Tecnoldgico de Celaya, Mexico. Used with permission.)...

Random Reactivity Uncertainties Affecting Shutdown Margins Uncertainties in calculations, input data, measurements, fuel loadings, basic constants, etc., must be taken into account in any estimate of core reactivity and shutdown margin calculations to ensure that the minimum criteria are met. Two types of uncertainties are considered, i.e., random uncertainties and systematic errors. The reactivity effects of random uncertainties, such as fuel loading tolerances, can be combined in a root mean square (RMS) fashion while the reactivity effects of systematic errors, such as core impurities, must be summed. 

The usefulness of equation 10 is limited for several reasons. The assumption of insignificant trace ion contribution to the conductivity is a poor one. There is no cancellation of trace cation and anion contributions as in the charge balance. Since the measured specific conductance is not a conservative quantity, mean values cannot be used. Major ion concentrations and measured specific conductance must be available for each sample. Random uncertainty in the measured values and the problem of small differences in relatively large numbers leads to large uncertainties in calculated pH values around 6.1 where... 

There are two problems with this picture. First a cluster is not really spherical it can actually be distorted from spherical (Fig. 5). Second, we do not know much about the sticking coefficient a,-, although it is generally assumed to be about 1.0 and independent of cluster size. Errors in the cluster area might increase c, from the value in Eq. (30) by as much as a factor of five or ten. Similarly if a, is actually 0.01 this would reduce the capture rate, and hence the nucleation rate, by a factor of 100. These errors in Eq. (30) for the capture rate will, of course, affect the nucleation rate. However, as we will see, these errors are insignificant compared with the uncertainties in calculating the equilibrium cluster concentrations, which we will examine below. 

The nuclear material contained and processed in a facility is stratified into items or batches that have similar physical and chemical characteristics. Grouping the material into strata simplifies verification and makes it possible to formulate the sampling plans needed to verify a material balance and to calculate its uncertainty. In calculating sampling plans, generally... 

---