Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 15 ppsx

3.3 Analytic Development of Reliability and Performance in Engineering Design 123 Computation: (only X) and (only Y)andG ⇒ (only Range (G, X, Y)) Flow [corners (Q, η , ω )] = (0.0375, 0.075, 0.45, 0.9) m 3 /min Flow [range (Q, η , ω )] = < flow 2.25 54 > m 3 /h Propagation result: Flow (Q ) = < all-parts only flow 2.25 54 > Elimination condition: (only X 1 ) and (only X 2 )andNot(X 1 ∩X 2 ) Subset interval: System requirement: X 1 = < flow 1.50 60 > m 3 /h Subset interval: X 2 = < flow 2.25 54 > m 3 /h Computation: (X 1 ∩X 2 ) = < flow 2.25 54 > m 3 /h Elimination result: Condition: Not (X 1 ∩X 2 ) ⇒true Description: With the labelled interval o f displacement between 0.5 ×10 −3 and 6 ×10 −3 cu- bic metre per revolution and the labelled interval of RPM in the interval of 75 to 150RPM, the pumpscan produce flows only in the interval of2 .25 to 54m 3 /h. The elimination condition is true in that the labelled interval of flow does no t meet the system requirement of: System requirement: X 1 = < flow 1.50 60 > m 3 /h Subset interval: X 2 = < flow 2.25 54 > m 3 /h 3.3.1.6 Labelled Interval Calculus in Designing for Reliability An approach to designing for reliability th at integrates functional failure as well as functional performance considerations so that a maximum safety margin is achieved with respect to all performance criteria is considered (Thompson et al. 1999). This approach has b een expanded to represent sets of systems functioning under sets of failure and performance intervals. The labelled interval calculus (LIC) formalises an approach for reasoning about these sets. The application of LIC in designing for reliability produces a design that has the highest possible safety margin with respect to intervals of performance values relating to specific system datasets. The most significant advantage of this expanded method is that, besides not having to rely on the propagation of single estimated values of failure data, it also does not have to rely on the determination of single values of maximum and minimum ac- ceptable limits of performance for each criterion. Instead, constraint propagation of intervals about sets of performance values is applied, making it possible to compute a multi-objective optimisation of conceptual design solution sets to different sets of performance intervals. 124 3 Reliability and Performance in Engineering Design Multi-objective optimisation of c onceptual design problems can be computed by applying LIC inference rules, which draw conclusions about the sets of systems under consideration to determine optimal solution sets to d ifferent intervals of per- formance values. Considering the performance limits represented diagram matically in Figs. 3.23, 3.24and 3.25, where an example of two performancelimits, one upper perfor mance limit, and one lower performance limit is given, the determination of datasets using LIC would include the following. a) Determination of a Data Point: Two Sets of L imit Intervals The proximity of actual performance to the minimum, nominal or maximum sets of limit intervals of performance for each performance criterion relates to a measure of the safety margin range. The data point x ij is the value closest to the nominal design condition that ap- proaches either minimu m or maximum limit interval. The value of x ij always lies in the range 0–10. Ideally, when the design condition is at the mid-range, then the data point is 10. A set of data points can thus be obtained for each system with re- spect to the performance parameters that are relevant to that system. In this case, the data point x ij approaching the maximum limit interval is the performance variable of temperature x ij = Max. Temp. T 1 −Nom. T High (×20) Max. Temp. T 1 −Min. Temp. T 2 (3.83) Given relationship: dataset: (Max. Temp. T 1 −Nom. T High)/(Max. Temp. T 1 −Min. Temp. T 2 ) ×20 where Max. Temp. T 1 = maximum performance interval Min. Temp. T 2 = minimum performance interval Nom. T High = nominal performance interval high Labelled intervals: Max. Temp. T 1 = < all-parts only T 1 t 1l t 1h > Min. Temp. T 2 = < all-parts only T 2 t 2l t 2h > Nom.T High = < all-parts only T H t Hl t Hh > where t 1l = lowest temperature value in interval of maximum performance interval. t 1h = highest temperature value in interval of maximum performance interval. t 2l = lowest temperature value in interval of minimum performance interval. t 2h = highest temperature value in interval of minimum performance interval. 3.3 Analytic Development of Reliability and Performance in Engineering Design 125 t Hl = lowest temperature value in interval of nominal performance interval high. t Hh = highest temperature value in interval of nominal performance interval high. Computation: propagation rule 1: (only X) and (only Y)andG ⇒ (only Range (G, X, Y)) x ij [corners (Max. Temp. T 1 ,Nom.T High, Min. Temp. T 2 )] =(t 1h −t Hl /t 1l −t 2h ) ×20 , (t 1h −t Hl /t 1l −t 2l ) ×20 , (t 1h −t Hl /t 1h −t 2h ) ×20 , (t 1h −t Hl /t 1h −t 2l ) ×20 , (t 1l −t Hl /t 1l −t 2h ) ×20 , (t 1l −t Hl /t 1l −t 2l ) ×20 , (t 1l −t Hl /t 1h −t 2h ) ×20 , (t 1l −t Hl /t 1h −t 2l ) ×20 , (t 1h −t Hh /t 1l −t 2h ) ×20 , (t 1h −t Hh /t 1l −t 2l ) ×20 , (t 1h −t Hh /t 1h −t 2h ) ×20 , (t 1h −t Hh /t 1h −t 2l ) ×20 , (t 1l −t Hh /t 1l −t 2h ) ×20 , (t 1l −t Hh /t 1l −t 2l ) ×20 , (t 1l −t Hh /t 1h −t 2h ) ×20 , (t 1l −t Hh /t 1h −t 2l ) ×20 , x ij [range (Max. Temp. T 1 ,Nom.T High, Min. Temp. T 2 )] =(t 1l −t Hh /t 1h −t 2l ) ×20 , (t 1h −t Hl /t 1l −t 2h ) ×20 Propagation result: x ij = < all-parts only x ij (t 1l −t Hh /t 1h −t 2l ) ×20 , (t 1h −t Hl /t 1l −t 2h ) ×20 > where x ij is dimensionless. Description: The generation of data points with respect to performance limits using the la- belled interval calculus, approaching the maximum limit interval. This is where the data point x ij approaching the maximum limit interval, with x ij in the range (Max.Temp.T 1 ,Nom.T High, Min. Temp. T 2 ), and the data point x ij being dimensionless,has a propagationresult equivalent to the following labelled interval: < all-parts only x ij (t 1l −t Hh /t 1h −t 2l )×20 , (t 1h −t Hl /t 1l −t 2h )×20 > ,which represents the relationship: x ij = Max. Temp. T 1 −Nom. T High (×20) Max. Temp. T 1 −Min. Temp. T 2 In the case of the data point x ij approaching the minimum limit interval,where the performance variable is temperature x ij = Nom. T Low−Min. Temp. T 2 (×20) Max. Temp. T 1 −Min. Temp. T 2 (3.84) 126 3 Reliability and Performance in Engineering Design Given relationship: dataset: (Max. Temp. T 1 −Nom. T High)/(Max. Temp. T 1 −Min. Temp. T 2 ) ×20 where Max. Temp. T 1 = maximum performance interval Min. Temp. T 2 = minimum performance interval Nom. T Low = nominal performance interval low Labelled intervals: Max. Temp. T 1 = < all-parts only T 1 t 1l t 1h > Min. Temp. T 2 = < all-parts only T 2 t 2l t 2h > Nom. T Low = < all-parts only T L t Ll t Lh > where t 1i = lowest temperature value in interval of maximum performance interval t 1h = highest temperature value in interval of maximum performance interval t 2l = lowest temperature value in interval of minimum performance interval t 2h = highest temperature value in interval of minimum performance interval t Ll = lowest temperature value in interval of nominal performance interval low t Lh = highest temperature value in interval of nominal performance interval low Computation: propagation rule 1 : (only X) and (only Y)andG ⇒ (only Range (G, X, Y)) x ij [corners (Max. Temp. T 1 ,Nom.T High, Min. Temp. T 2 )] =(t Lh −t 2l /t 1l −t 2h ) ×20 , (t Lh −t 2l /t 1l −t 2l ) ×20 , (t Lh −t 2l /t 1h −t 2h ) ×20 , (t Lh −t 2l /t 1h −t 2l ) ×20 , (t Ll −t 2l /t 1l −t 2h ) ×20 , (t Ll −t 2l /t 1l −t 2l ) ×20 , (t Ll −t 2l /t 1h −t 2h ) ×20 , (t Ll −t 2l /t 1h −t 2l ) ×20 , (t Lh −t 2h /t 1l −t 2h ) ×20 , (t Lh −t 2h /t 1l −t 2l ) ×20 , (t Lh −t 2h /t 1h −t 2h ) ×20 , (t Lh −t 2h /t 1h −t 2l ) ×20 , (t Ll −t 2h /t 1l −t 2h ) ×20 , (t Ll −t 2h /t 1l −t 2l ) ×20 , (t Ll −t 2h /t 1h −t 2h ) ×20 , (t Ll −t 2h /t 1h −t 2l ) ×20 , x ij [range (Max. Temp. T 1 ,Nom.T High, Min. Temp. T 2 )] =(t Ll −t 2h /t 1h −t 2l ) ×20 , (t Lh −t 2l /t 1l −t 2h ) ×20 3.3 Analytic Development of Reliability and Performance in Engineering Design 127 Propagation result: x ij = < all-parts only x ij (t Ll −t 2h /t 1h −t 2l ) ×20 , (t Lh −t 2l /t 1l −t 2h ) ×20 > where x ij is dimensionless. Description: The generation of data points with respect to performance limits using the la- belled interval calculus, in the case of the data point x ij approachingthe minimum limit interval, with x ij in the range (Max. Temp. T 1 ,Nom.T High, Min. Temp. T 2 ), and x ij dimensionless, has a propagation result equivalent to the following labelled interval: < all-parts only x ij (t Ll −t 2h /t 1h −t 2l ) ×20 , (t Lh −t 2l /t 1l −t 2h ) ×20 > which represents the relationship: x ij = Nom. T Low−Min. Temp. T 2 (×20) Max. Temp. T 1 −Min. Temp. T 2 b) Determination of a Data Point: One Upper Limit I nterval If there is one operating limit set only, then the data point is obtained as shown in Figs. 3.24 and 3.25, where the upper or lower limit is known. A set of data points can be obtained for each system with respect to the performance parameters that are relevant to that system. In the case of the data point x ij approaching the upper limit interval x ij = Highest Stress Level−Nominal Stress Level (×10) Highest Stress Level−Lowest Stress Est. (3.85) Given relationship: dataset: (HSL−NSL)/(HSL−LSL) ×10 Labelled intervals: HSI = highest stress interval < all-parts only HSI s 1l s 1h > LSI = lowest stress interval < all-parts only LSI s 2l s 2h > NSI = nominal stress interval < all-parts only NSI s Hl s Hh > where: s 1l = lowest stress value in interval of highest stress interval s 1h = highest stress value in interval of highest stress interval s 2l = lowest stress value in interval of lowest stress interval s 2h = highest stress value in interval of lowest stress interval s Hl = lowest stress value in interval of nominal stress interval s Hh = highest stress value in interval of nominal stress interval 128 3 Reliability and Performance in Engineering Design Computation: propagation rule 1 : (only X) and (only Y)andG ⇒(only Range (G, X, Y)) x ij [corners (HSL, NSL, LSL)] =(s 1h −s Hl /s 1l −s 2h ) ×10 , (s 1h −s Hl /s 1l −s 2l ) ×10 , (s 1h −s Hl /s 1h −s 2h ) ×10 , (s 1h −s Hl /s 1h −s 2l ) ×10 , (s 1l −s Hl /s 1l −s 2h ) ×10 , (s 1l −s Hl /s 1l −s 2l ) ×10 , (s 1l −s Hl /s 1h −s 2h ) ×10 , (s 1l −s Hl /s 1h −s 2l ) ×10 , (s 1h −s Hh /s 1l −s 2h ) ×10 , (s 1h −s Hh /s 1l −s 2l ) ×10 , (s 1h −s Hh /s 1h −s 2h ) ×10 , (s 1h −s Hh /s 1h −s 2l ) ×10 , (s 1l −s Hh /s 1l −s 2h ) ×10 , (s 1l −s Hh /s 1l −s 2l ) ×10 , (s 1l −s Hh /s 1h −s 2h ) ×10 , (s 1l −s Hh /s 1h −s 2l ) ×10 , x ij [range (HSL, NSL, LSL)] =(s 1l −s Hh /s 1h −s 2l ) ×10 , (s 1h −s Hl /s 1l −s 2h ) ×10 Propagation result: x ij = < all-parts only x ij (s 1l −s Hh /s 1h −s 2l ) ×10 , (s 1h −s Hl /s 1l −s 2h ) ×10 > where x ij is dimensionless. Description: The data point x ij approachingthe upperlimit interval, with x ij in the range (High Stress Level, Nominal Stress Level, Lowest Stress Level), and x ij dimensionless, has a propagation result equivalent to the following labelled interval: < all-parts only x ij (s Ll −s 2h /s 1h −s 2l ) ×20 , (s Lh −s 2l /s 1l −s 2h ) ×20 > , which represents the relationship: x ij = Highest Stress Level−Nominal Stress Level (×10) Highest Stress Level−Lowest Stress Est. c) Determination o f a Data Point: One Lower Limit I nterval In the case of the data point x ij approaching the lower limit interval x ij = Nominal Capacity−Min. Capacity Level (×10) Max. Capacity Est.−Min. Capacity Level (3.86) Given relationship: dataset: (Nom. Cap. L−Min. Cap. L)/(Max. Cap. L−Min. Cap. L) ×10 where Max. Cap. C 1 = m a ximum capacity interval Min. Cap. C 2 = m inimum capacity interval Nom. Cap. C L = nominal capacity interval low 3.3 Analytic Development of Reliability and Performance in Engineering Design 129 Labelled intervals: Max. Cap. C 1 = < all-parts only C 1 c 1l c 1h > Min. Cap. C 2 = < all-parts only C 2 c 2l c 2h > Nom. Cap. C L = < all-parts only C L c Ll c Lh > where c 1l = lowest capacity value in interval of maximum capacity interval c 1h = highest capacity value in interval of maximum capacity interval c 2l = lowest capacity value in interval of minimum capacity interval c 2h = highest capacity value in interval of minimum capacity interval c Ll = lowest capacity value in interval of nominal capacity interval low c Lh = highest capacity value in interval of nominal capacity interval low Computation: propagation rule 1: (only X) and (only Y)andG ⇒ (only Range (G, X, Y)) x ij [corners (Max. Cap. Min. Cap. C 2 ,Nom.Cap.C L )] =(c Lh −c 2l /c 1l −c 2h ) ×10 , (c Lh −c 2l /c 1l −c 2l ) ×10 , (c Lh −c 2l /c 1h −c 2h ) ×10 , (c Lh −c 2l /c 1h −c 2l ) ×10 , (c Ll −c 2l /c 1l −c 2h ) ×10 , (c Ll −c 2l /c 1l −c 2l ) ×10 , (c Ll −c 2l /c 1h −c 2h ) ×10 , (c Ll −c 2l /c 1h −c 2l ) ×10 , (c Lh −c 2h /c 1l −c 2h ) ×10 , (c Lh −c 2h /c 1l −c 2l ) ×10 , (c Lh −c 2h /c 1h −c 2h ) ×10 , (c Lh −c 2h /c 1h −c 2l ) ×10 , (c Ll −c 2h /c 1l −c 2h ) ×10 , (c Ll −c 2h /c 1l −c 2l ) ×10 , (c Ll −c 2h /c 1h −c 2h ) ×10 , (c Ll −c 2h /c 1h −c 2l ) ×10 , x ij [range (Max. Cap. Min. Cap. C 2 ,Nom.Cap.C L )] =(c Ll −c 2h /c 1h −c 2l ) ×10 , (c Lh −c 2l /c 1l −c 2h ) ×10 Propagation result: x ij = < all-parts only x ij (c Ll −c 2h /c 1h −c 2l ) ×10 , (c Lh −c 2l /c 1l −c 2h ) ×10 > where x ij is dimensionless. Description: The generation of data points with respect to performance limits using the la- belled interval calculus for the lower limit interval is the following: 130 3 Reliability and Performance in Engineering Design The data pointx ij approachingthe lower limitinterval, with x ij in the range (Max. Capacity Level, Min. Capacity Level, Nom. Capacity Level), and x ij dimension- less, has a propagation result equivalent to the following labelled interval: < all-parts only x ij (c Ll −c 2h /c 1h −c 2l ) ×10 , (c Lh −c 2l /c 1l −c 2h ) ×10 > with x ij in the range (Max. Cap. Min. Cap. C 2 ,Nom.Cap.C L ), representing the relationship: x ij = Nominal Capacity−Min. Capacity Level(×10) Max. Capacity Est.−Min. Capacity Level d) Analysis of the Interval Matrix In Fig. 3.26, the performance measures of each system of a process are described in matrix form containing data points relating to pr ocess systems and single pa- rameters that describe their performance. The matrix can be analysed by rows and columns in order to evaluate the performance characteristics of the process. Each data point of x ij refers to a single parameter. Similar ly, in the expande d method using labelled interval calculus (LIC), the performance measures of each system of a process are described in an interval matrix form, containing datasets relatin g to systems and labelled intervals that describe their performance. Each row of the in- terval matrix reveals whether the process has a consistent safety margin with respect to a specific set o f p erformance values. A parameter performance index, PPI, can be calculated for each row PPI = n  n ∑ j= 1 1  x ij  −1 (3.87) where n is th e number of systems in row i. The calculation of PPI is accomplished using LIC inference rules that draw con- clusions about the system datasets of each matrix row under consideration. The numerical value of PPI lies in the range 0–10, irrespective of the number of datasets in each row (i.e. the number of process systems). A comparison of PPIs can be made to judge whether specific performance criteria, such as reliability, are acceptable. Similarly, a system performance index, SPI, can be calculated for each column as SPI = m  m ∑ i=1 1  x ij  −1 (3.88) where m is the number of parameters in column i. The calculation of SPI is accomplished using LIC inference rules that draw con- clusions about performance labelled intervals of each matrix column under con- sideration. The numerical value of SPI also lies in the range 0–10, irrespective of the number of labelled in tervals in each column (i.e. the number of performance 3.3 Analytic Development of Reliability and Performance in Engineering Design 131 parameters). A comparison of SPIs can be made to assess whether there is accept- able performance with respect to any performance criteria of a specific system. Finally, an overall performance index, OPI, can be calculated (Eq. 3.89). The numerical value of OPI lies in the range 0–100 and can be indicated as a percentage value. OPI = 1 mn  m ∑ i=1 n ∑ j= 1 (PPI)(SPI)  (3.89) where m is the number of performance parameters, and n is the number of systems. Description of Example Acidic gases, such as sulphur dioxide, are removed from the combustion gas emis- sions of a non-ferrous metal smelter by passing these through a reverse jet scrub- ber. A reverse jet scrubber consists of a scrubber vessel containing jet-spray nozzles adapted to spray,under high pressure, a caustic scrubbing liquid counter to the high- velocity combustion gas stream emitted by the smelter, whereby the combustion gas stream is scrubbed and a clear gas stream is recovered downstream. The reverse jet scrubber consists of a scrubber vessel and a subset of three centrifugal pumps in parallel, any two of which are continually operational, with the following labelled intervals for the specific performance parameters (Tables 3.10 and 3.11): Propagation result: x ij = < all-parts only x ij (x 1l −x Hh /x 1h −x 2l ) ×10 , (x 1h −x Hl /x 1l −x 2h ) ×10 > Table 3.10 Labelled intervals for specific performance parameters Parameters Vessel Pump 1 Pump 2 Pump 3 Max. flow < 65 75 ><55 60 ><55 60 ><65 70 > Min. flow < 30 35 ><20 25 ><20 25 ><30 35 > Nom. flow < 50 60 ><40 50 ><40 50 ><50 60 > Max. pressure < 10000 12500 ><8500 10000 ><8500 10000 ><12500 15000 > Min. pressure < 1000 1500 ><1000 1250 ><1000 1250 ><2000 2500 > Nom. pressure < 5000 7500 ><5000 6500 ><5000 6500 ><7500 10000 > Max. temp. < 80 85 ><85 90 ><85 90 ><80 85 > Min. temp. < 60 65 ><60 65 ><60 65 ><55 60 > Nom. temp. < 70 75 ><75 80 ><75 80 ><70 75 > Table 3.11 Parameter interval matrix Parameters Vessel Pump 1 Pump 2 Pump 3 Flow (m 3 /h) < 1.18.3 ><1.36.7 ><1.36.7 ><1.18.3 > Pressure (kPa) < 2.28.8 ><2.26.9 ><2.26.9 ><1.97.5 > Temp. ( ◦ C) < 2.010.0 ><1.77.5 ><1.77.5 ><1.75.0 > 132 3 Reliability and Performance in Engineering Design Labelled intervals—flow: Vessel interval: = < all-parts only x ij 1.18.3 > Pump1interval:= < all-parts only x ij 1.36.7 > Pump2interval:= < all-parts only x ij 1.36.7 > Pump3interval:= < all-parts only x ij 1.18.3 > Labelled intervals—pressure: Vessel interval: = < all-parts only x ij 2.28.8 > Pump1interval:= < all-parts only x ij 2.26.9 > Pump2interval:= < all-parts only x ij 2.26.9 > Pump3interval:= < all-parts only x ij 1.97.5 > Labelled intervals—temperature: Vessel interval: = < all-parts only x ij 2.010.0 > Pump1interval:= < all-parts only x ij 1.77.5 > Pump2interval:= < all-parts only x ij 1.77.5 > Pump3interval:= < all-parts only x ij 1.75.0 > The parameter performance index, PPI, can be calculated for each row PPI = n  n ∑ j= 1 1  x ij  −1 (3.90) where n is th e number of systems in row i. Labelled intervals: Flow (m 3 /h) PPI = < all-parts only PPI 1.27.4 > Pressure (kPa) PPI = < all-parts only PPI 2.17.5 > Temp. ( ◦ C) PPI = < all-parts only PPI 1.87.1 > The system performance index, SPI, can be calculated for each column SPI = m  m ∑ i=1 1  x ij  −1 (3.91) where m is the number of parameters in column i. Labelled intervals: Vessel SPI = < all-parts only 1.69.0 > Pump 1 SPI = < all-parts only 1.77.0 > Pump 2 SPI = < all-parts only 1.77.0 > Pump 3 SPI = < all-parts only 1.56.6 > Description: The parameter performance index, PPI,andthesystem performance index, SPI, indicate whether there is acceptable overall performance of the operational pa- rameters (PPI), and what contribution an item makes to the overall effectiveness of the system (SPI). . temperature value in interval of minimum performance interval. 3.3 Analytic Development of Reliability and Performance in Engineering Design 125 t Hl = lowest temperature value in interval of nominal performance. lowest stress value in interval of nominal stress interval s Hh = highest stress value in interval of nominal stress interval 128 3 Reliability and Performance in Engineering Design Computation:. value in interval of minimum performance interval t 2h = highest temperature value in interval of minimum performance interval t Ll = lowest temperature value in interval of nominal performance interval