Integration errors

Chromatographic Integration Methods, Royal Society of Chemistry, London, 2nd edn, 1998. K. Ogan, in Quantitative Analysis using Chromatographic Techniques, E. Katz, ed., John Wiley Sons, 

In many cases integration errors cannot be compensated for by injection of standards and determination of calibration factors because a standard mixture is usually less complex than the sample. 

Electronic integrators and data systems are of such indispensable importance that possible errors as a result of their use can be overlooked. In order to become familiar with these problems it is helpful to read the easy to understand book by Dyson or some review papers. Different types of error can occur, the most important ones being presented in Fig. 19.8. 

Chromatographic Integration Methods, Royal Society of Chemistry, London, 2nd ed., 1998. 

In many cases integration errors cannot be compensated for by injection of 

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For a PI or PID controller, the integrated deviation—better known as the integrated error IE—is related to the controller settings ... 

Feedforward control can also be applied by multiplying the liquid flow measurement—after dynamic compensation—by the output of the temperature controller, the result used to set steam flow in cascade. Feedforward is capable of a reduction in integrated error as much as a hundredfold but requires the use of a steam-flow loop and dynamic compensator to approach this. 

Usually, simplified representations of the data are used to obtain preliminary structures. Thus, lower and upper bounds on the interproton distances are estimated from the NOE intensity [10], using appropriate reference distances for calibration. The bounds should include the estimates of the cumulative error due to all sources such as peak integration errors, spin diffusion, and internal dynamics. 

Implementing an Integrated Error and Process Safety Management System at the Plant... 

The marching-ahead technique systematically overestimates when component A is a reactant since the rate is evaluated at the old concentrations where a and 0t A are higher. This creates a systematic error similar to the numerical integration error shown in Figure 2.1. The error can be dramatically reduced by the use of more sophisticated numerical techniques. It can also be reduced by the simple expedient of reducing At and repeating the calculation. 

Integral error criteria are ideally suited to simulation applications since only one additional program statement is required for the simulation. The optimal control parameters Kp, Tj and Xq can be then found at minimal ITAE. For this, it is useful to be able to apply the available optimisation tools implemented in such programs as MATLAB, SIMUSOLV and ESL. 

Equation 10.25 is of the form AAx=b which can be solved for Ax 1 and thus x0+l)(t .,) is obtained. Normally, we converge to x(tI+]) in very few iterations. If however, convergence is not achieved or the integration error tolerances are not satisfied, the time-step is reduced and the computations are repeated. 

In real life, we expect probable simultaneous reference and disturbance inputs. As far as analysis goes, the mathematics is much simpler if we consider one case at a time. In addition, either case shares the same closed-loop characteristic polynomial. Hence they should also share the same stability and dynamic response characteristics. Later when we talk about integral error criteria in controller design, there are minor differences, but not sufficient to justify analyzing a problem with simultaneous reference and load inputs. 

Hummer, G., Fast-growth thermodynamic integration error and efficiency analysis, J. Chem. Phys. 2001, 114. 7330-7337... 

A warning It is possible to obtain results from a program that at first glance seems OK. Always make sure that the same results are obtained when DT is reduced by a factor of 10 or when a different integration method is used. Plotting all the variables may reveal oscillations that indicate integration errors. These may not be detectable on plots of the other variables. 

The hematite with adsorbed Co-57 or Sb-119 along with the solution was subjected to emission Mossbauer measurement at 24 1°C with the experimental setup shown in Figure 2. The absorber, Fe-57-enriched potassium ferrocyanide (0.5 mg Fe-57/cm2) or barium stannate (0.9 mg Sn-119/cm2), was driven by a Hanger 700-series Mossbauer spectrometer connected to a Tracor-Northern TN-7200 multi-channel analyzer. The Mosssbauer gamma-rays of Co-57 and Sb-119 were detected respectively with a Kr(+3% carbon dioxide)-filled proportional counter and with a 2 mm-thick Nal(Tl) scintillation counter through 65 pm-thick Pd critical absorber for Sn K X-rays. The integral errors in the relative velocity were estimated to be of the order of 0.05 mm/s by repeated calibration measurements using standard absorbers. 

Sample concentration-. Higher sample concentrations or larger sample volumes reduce integration errors and thus increase precision. ... 

In addition to stability considerations, the order of the approximation error is also a function of the system index. From the results of Brenan and Petzold (1987), systems of equations of higher index can be considered simply by choosing the appropriate IRK method with the appropriate integration error constraints. Based on error and stability considerations, Logsdon and Biegler (1989) concluded that minimum order requirements for collocation methods are the following ... 

A long-range fluctuation contribution that describes the longer-term trends in the process. The new term process integration error (PIE2) covers what used to be called the time fluctuation error (TEE) in earlier literature. 

Next we turn to the inference of cooling history. The length of the concentration profile in each phase is a rough indication of (jDdf) = (Dot), where Do is calculated using Tq estimated from the thermometry calculation. If can be estimated, then x, Xc and cooling rate q may be estimated. However, because the interface concentration varies with time (due to the dependence of the equilibrium constants between the two phases, and a, on temperature), the concentration profile in each phase is not a simple error function, and often may not have an analytical solution. Suppose the surface concentration is a linear function of time, the diffusion profile would be an integrated error function i erfc[x/(4/Ddf) ] (Appendix A3.2.3b). Then the mid-concentration distance would occur at... 

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