10 pág.

# Practical applications of sampling theory

Pré-visualização6 páginas

had to be treated as separate strata. 4.2.2.1. Design requirements. \ufffd Material balance for sulfur required annually \ufffd A two-man team with portable instruments available for atmospheric emission control measurements \ufffd To avoid too much time spent in transporting and setting up the instruments, a one-week measurement period is carried out at each location, where the team is working; that is, 52 periods/year are available and should be allocated optimally \ufffd For optimization, the emission sources were grouped into six groups, which could be measured during 1 week from one station in the field 4.2.2.2. Acquisition of basic data. Existing records sup- plemented with new experiments were used to estimate: Fig. 7. To estimate the uncertainty of the annual sulfur balance in a pulp factory, the standard deviations of the means of several material streams have to be estimated. The streams have different sizes, contain sulfur in many different compounds, and in many streams, high variability is characteristic to the sulfur containing compounds. This is especially the case for atmospheric emissions.). \ufffd Relative contribution (size) of all major emission sources Fig. 8. Heterogeneity (relative variation about the mean) of SO2 in emissions from a soda recovery boiler during a measurement period. Table 1 Relative sizes, relatives standard deviation of one sample/measurement period, allocation of 32 samples (measurement periods) that can be made during one week, and relative standard deviations of the mean of 1 week in emission group #6 Stratum No Wi sri (%) ni sr(x¯) (%) 1 0.143 30 5 13.4 2 0.285 25 9 8.3 3 0.002 40 1 40 4 0.285 25 9 8.3 5 0.285 25 9 8.3 Total 1.000 32 srðx¯¯Þ ¼ 4:5% P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u201394 93 to the total emission of sulfur compounds \ufffd Variance estimates for the different emission sources (e.g., from variogram or from Analysis of Variance). As an example, Fig. 8 shows the SO2 emissions from the soda recovery boiler as the process heterogeneity (relative variation about the process mean). As can be seen, a high variability is characteristic to emissions from this source. In the peaks, the SO2 concentration is three times higher than the process average. From the heterogeneity values, variogram (Fig. 9) can be calculated for the process. Fig. 9 also shows the relative standard deviation estimates for the sampling error in this process stream as function of the sampling interval both for the systematic and the stratified Fig. 9. Variogram calculated from the heterogeneities of Fig. 8 (upper part) and relative standard deviation estimates for systematic and stratified sample selection as function of the sampling interval calculated from the variogram (lower part). sample selection. The variogram and standard deviation estimates were calculated by using Gy\u2019s method. With the standard deviation estimates, the uncertainty of the process mean during the measurement period can be estimated. This measurement plan can be treated as a two- level hierarchical sampling plan, where one sample is analyzed from each of the N1 strata. By substituting N1 = n1, n2 = 1 (and N2Hn2) and N3 = n3 = 0 in Eq. (8a), the standard deviation of the mean of the measurement period is obtained as r2x¯ ¼ N1 \ufffd n1 N1 \ufffd 1 r21 n1 þ N2 \ufffd n2 N2 \ufffd 1 r22 n1n2 c s22 n1 ; where s2 2 is the standard deviation estimate from Fig. 9 for the used sampling frequency, and n1 is the number of samples analyzed. 4.2.2.3. Optimization procedure. Optimization was car- ried out at two levels: (1) For each group having more than one source, the analytical resources were optimally allocated within the one-week (5 days) measurement period between the different sources (an example is given in Table 1). (2) Of the 52 measurement weeks, two were allocated for the insignificant sources, where only occasional measurements need to be carried out, and 50 weeks were allocated to the six most important groups. The results of the final optimization are given in Table 2. Table 2 Relative sizes, total relatives standard deviation between the mean of 1- week measurement period, and allocation of 50 measurement weeks between the different emission source groups Emission group no. Wi (%) sri (%) ni 1 0.14 14.6 10 2 0.18 20.8 17 3 0.04 14.8 3 4 0.02 7.9 1 5 0.02 10.9 1 6 0.60 6.4 18 Standard deviation of the annual mean of sulfur emission, srðx¯¯Þ ¼ 1:5% Table 2 shows that regardless of the high heterogeneity of sulfur in gaseous emissions, the mean can be estimated fairly accurate at an annual level. When the emission measurements were carried out approximately according to the optimized plan, the unaccounted sulfur in the annual material balance was only about 1%, which is in a good agreement with the theoretical calculation considering the complexity of the process. 5. Conclusions Sampling theory can, and should, be applied in all steps directly to a paper factory through a pipeline. The pulp factory had installed a process analyzer to the pipeline and the amount of pulp pumped to the paper factory\u2014and the bill for sold pulp\u2014was based on this measurement. When the discrepancy in the mass-balances of paper produced and pulp received was noticed, the sampling and analytical chain was carefully checked. It turned out that partly due to the sampling problems and partly due to the calibration problems of the analyzer, the estimated amount of pulp could be 10% too high. As the paper factory consumed about 80000 tonne pulp per year, the analytical error was really expensive. By improving the sampling and calibra- tion procedure, the systematic errors could be removed and [1] P.M. Gy, Sampling of Particulate Materials, Theory and Practice, New York, 1986. [7] W.G. Cochran, Sampling Techniques, Third edition, Wiley, New York, P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u20139494 of analytical procedures, from the planning to the analytical measurements. At the moment, it is largely neglected. A large amount of analytical resources is devoted for quality control and environmental emission estimation in industries. At national levels, large programs are devoted, e.g., to monitor the state of environment and to guarantee the quality and safety of food. Only seldom that the data quality objectives are adequately defined before the sampling campaigns, and consequently, the sampling theory is not utilized when these campaigns are designed. Often, the sampling plans have just evolved and their performance has never been critically audited. This leaves a lot of room for optimization in this field. Before designing a sampling and analytical plan, one should carefully consider what is the uncertainty level tolerated by the user of the results. If the ambition level is set too high, the investigation will be unnecessarily expensive. In general, in the cost\u2013benefit relationship considerations, the following rule applies: if the total standard deviation is cut to half, the cost will increase four times, and if only one quarter of original standard deviation can be tolerated, the cost will be 16 times higher, etc. But, what is important, this relationship only holds if the analytical procedure has been optimized. If the resour- ces are not optimally allocated, the required uncertainty level may not be achieved at all. On the other hand, optimization of existing procedures may become cheaper and still give more reliable result than the original proce- dure. An example: a pulp factory pumped the pulp slurry 1977. [8] P. Minkkinen, SAMPEX\u2014a computer program for solving sampling problems, Chemolab 7 (1989) 189\u2013194. Elsevier, Amsterdam, 1982. [2] P.M. Gy, Sampling of Heterogeneous and Dynamic Material Systems, Elsevier, Amsterdam, 1992. [3] P.M. Gy, Sampling for Analytical Purposes, Wiley, Chichester, 1998. [4] F.F. Pitard, Pierre Gy\u2019s Sampling Theory and Sampling Practice,