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# Fuzzy Evaluation of stream sample reability

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Manual The electric drive is the only drive likely to achieve constant speed of the cutter across a flowing stream of particulate material (Pitard 1993). 3.3. Cutter Layout Idle positions of the cutter should be outside the stream to permit the motor to accelerate to its cruising speed before the cutter attains the stream. When one idle position of the cutter is in the stream, as displayed in Figure 4, a part of the stream enters the cutter all the time. The shaded area illustrates the material that is collected but does not belong to the increment. This situation is incorrect. FUZZY EVALUATION OF STREAM SAMPLE RELIABILITY 285 In addition, the stream should fall exactly in the center of the cutter. When the stream thickness is larger than the cutter length, the cutter lay- out is incorrect. Hence, a part of the stream falls outside the cutter tra- jectory, as shown in Figure 5. The shaded area illustrates the material belonging to the increment that is not collected. Furthermore, the minimum vertical distance u (see Figure 6) between the liberation point of the stream and the plane described by the cutter edges during sampling should be respected (Pitard 1993). This prevents accidental obstruction between the cutter and the point of dis- charge of the stream. 3.4. Cutter Path. The condition of cutter path correctness is that the cutter path is perpendicular to the direction of the stream. For instance, Figure 4. Incorrect cutter layout with idle position in the stream. Figure 5. Incorrect cutter layout with stream thickness larger than cutter length. 286 C. KETATA AND M. C. ROCKWELL if the cutter path is parallel to the stream direction, the sampling operation is considered to be performed by the scheme: taking part of the stream all of the time. This is always incorrect and is illustrated in Figure 7. The shaded area represents the material collected by the cutter. 4. FUZZY SETS The following fuzzy sets correspond to the sampling factors\u2019 reliability. These fuzzy sets depend on the sampling situation characteristics such as material heterogeneity. Figure 6. Vertical distance u between the liberation point of the stream and the plane described by the cutter edges. Figure 7. Incorrect cutter path with taking part of the stream all of the time. FUZZY EVALUATION OF STREAM SAMPLE RELIABILITY 287 4.1. Cutter Geometry Here, the cutter is assumed to be resistant to violent impact of large frag- ments. In this way, the cutter geometry becomes incorrect if it is obstructed by sticky material such as fines. Figure 8 illustrates an example of possible membership function of the fuzzy set \u2018\u2018very reliable cutter geometry\u2019\u2019: eAA ¼ \u2018\u2018very reliable cutter geometry\u2019\u2019 ¼ fðv; l ~AAðvÞÞjv 2 ½0; 1\ufffdg ð1Þ In this expression, v denotes the area fraction of the cutter opening given by: v ¼ AE A ð2Þ where AE represents the effective cutter area and A is the total cutter area as shown in Figure 3. The corresponding membership function equals: leAAðvÞ ¼ v; 0 \ufffd v \ufffd 1 ð3Þ 4.2. Cutter Speed The faulty cutter speed results when the cutter speed is not constant dur- ing sampling. Figure 9 exhibits an example of possible membership func- tion of the fuzzy set \u2018\u2018very reliable cutter speed\u2019\u2019: eBB ¼ \u2018\u2018very reliable cutter speed ¼ fðx;leBðxÞÞjx 2 ½0; 1\ufffdg ð4Þ Figure 8. Example of fuzzy set \u2018\u2018very reliable cutter geometry.\u2019\u2019 288 C. KETATA AND M. C. ROCKWELL where x is the constant-speed length fraction of the cutter given by: x ¼ LA LT ð5Þ where LA symbolizes the actual cutter path length that is perpendicular to the stream direction and equal to the length of constant speed and LT is the stream width. The corresponding membership function is: leBðxÞ ¼ x; 0 \ufffd x \ufffd 1 ð6Þ 4.3. Cutter Layout Here, the minimum vertical distance u, which prevents accidental obstruction between the cutter and the point of stream discharge, is assumed to be respected. In addition, the cutter is presumed to have its idle positions completely outside the flowing stream. Thus, the faulty layout results when the stream thickness is larger than the cutter length. Figure 10 displays an example of possible membership function of the fuzzy set \u2018\u2018very reliable cutter layout\u2019\u2019: eCC ¼ \u2018\u2018very reliable cutter layout\u2019\u2019 ¼ fðy; leCðyÞÞjy 2 ½0;þ1\ufffdg ð7Þ where y is the cutter length fraction given by: y ¼ L l ð8Þ Figure 9. Example of fuzzy set \u2018\u2018very reliable cutter speed.\u2019\u2019 FUZZY EVALUATION OF STREAM SAMPLE RELIABILITY 289 where L and l represent the stream thickness and the cutter length, respectively (see Figure 5). The related membership function is written as leCðyÞ ¼ 1; 0 \ufffd y \ufffd 11=y; y \ufffd 1 \ufffd ð9Þ 4.4. Cutter Path The incorrect cutter path results when the cutter deviates from the cor- rect route that is perpendicular to the stream direction. Figure 11 shows an example of possible membership function of the fuzzy set \u2018\u2018very Figure 10. Example of fuzzy set \u2018\u2018very reliable cutter layout.\u2019\u2019 Figure 11. Example of fuzzy set \u2018\u2018very reliable cutter path.\u2019\u2019 290 C. KETATA AND M. C. ROCKWELL reliable cutter path\u2019\u2019: eDD ¼ \u2018\u2018very reliable cutter path\u2019\u2019 ¼ fðz; leDðzÞÞjz 2 ½0\ufffd; 90\ufffd\ufffdg ð10Þ where z is the cutter deviation angle, in degrees, from the correct cutter path that is perpendicular to the stream (see Figure 7). The related mem- bership function is described by: leDðzÞ ¼ cosðzÞ; 0 \ufffd z \ufffd 90\ufffd ð11Þ 5. FUZZY EVALUATION OF SAMPLE RELIABILITY The sample reliability is assumed to be influenced by the sampling factors: 1) Cutter geometry 2) Cutter Speed 3) Cutter layout 4) Cutter path To assess the sample reliability, the compound influence of the sampling factors should be evaluated. This situation is essentially a multi-criterion decision-making problem. The sampling factors are put into the set C: C ¼ ½cutter geometry, cutter speed, cutter layout, cutter path\ufffd: ð12Þ The sample reliability is described by five indexes, namely very reliable, r1; reliable, r2; adequate, r3; doubtful, r4; and very doubtful, r5. These indexes are represented by index set R: R ¼ ½r1; r2; r3; r4; r5\ufffd ð13Þ Thus, the fuzzy relation eRR is established: eRR ¼ \u2018\u2018 ci is rj \u2019\u2019 ð14Þ where i ¼ 1; . . . ; 4 and j ¼ 1; . . . ; 5. Then, eRR is defined by the matrix: eRR ¼ r11 r12 r13 r14 r15 r21 r22 r23 r24 r25 r31 r32 r33 r34 r35 r41 r42 r43 r44 r45 2 664 3 775 ð15Þ FUZZY EVALUATION OF STREAM SAMPLE RELIABILITY 291 where rij 2 ½0; 1\ufffd is the membership grade of the pair (ci,rj) given by: rij ¼ leRðci; rjÞ: ð16Þ Now, the degree of sampling factor importance is represented by the fuzzy set eII : eII ¼ ½i1; i2; i3; i4\ufffd ð17Þ where ik is the membership grade of factor ck. The quantifying scales of the perceived importance of each sampling factor can be determined by using the factor importance ruler shown in Figure 12. Consequently, the overall assessment result is obtained by: eWW ¼ eJJ \ufffd eRR ¼ ½ j1; j2; j3; j4\ufffd \ufffd r11 r12 r13 r14 r15 r21 r22 r23 r24 r25 r31 r32 r33 r34 r35 r41 r42 r43 r44 r45 2 6664 3 7775 ¼ w1;w2;w3;w4;w5½ \ufffd ð18Þ where \ufffd stands for product and jk equals: jk ¼ ikP4 l¼1 il ; k ¼ 1; . . . ; 4: ð19Þ wi is the membership grade of sample reliability index i. The sample reliability is described by the index i with the highest value wi. To illustrate the above formulae, a stream sampling situation is simu- lated. Its membership grade matrix eRR is given by: eRR ¼ 0:8 0:2 0 0 0 0:2 0:4 0:3 0:1 0 0:6 0:3 0:1 0 0 0:2 0:3 0:4 0:1 0 2 664 3 775: ð20Þ Figure 12. Factor importance ruler. 292 C. KETATA AND M. C. ROCKWELL In this case, all the sampling factors are extremely important. Therefore, the factor importance set eII is obtained as eII ¼ ½1; 1; 1; 1\ufffd: ð21Þ Then, the following judgment result is acquired: eWW ¼ eJJ \ufffd eRR ¼ ½0:45; 0:3; 0:2; 0:05; 0\ufffd: ð22Þ From the fuzzy set eWW , it appears that the membership grade for index 2, reliable, is the maximum. This indicates that the sample is very reliable with a degree of 45%. 6. CONCLUSION This article describes