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

The elimination condition is true in that the labelled interval of flow does not meet the system requirement of: System requirement: X1= < flow 1.50 60 > m3/h Subset interval: X2= < flow

(only X ) and (only Y ) and G ⇒ (only Range (G, X, Y ))

Flow [corners (Q,η,ω)]= (0.0375, 0.075, 0.45, 0.9) m3/min

Flow [range (Q,η,ω)] = < flow 2.25 54 > m3/h

Propagation result: Flow (Q) = < all-parts only flow 2.25 54 >

Elimination condition:

(only X1) and (only X2) and Not (X1∩ X2)

Subset interval:

System requirement: X1= < flow 1.50 60 > m3/h

Subset interval: X2= < flow 2.25 54 > m3/h

Computation:

(X1∩ X2)= < flow 2.25 54 > m3/h

Elimination result:

Condition: Not (X1∩ X2)⇒true

Description:

With the labelled interval of displacement between 0.5 ×10 −3and 6×10 −3

cu-bic metre per revolution and the labelled interval of RPM in the interval of 75

to 150 RPM, the pumps can produce flows only in the interval of 2.25 to 54 m3/h.

The elimination condition is true in that the labelled interval of flow does not meet the system requirement of:

System requirement: X1= < flow 1.50 60 > m3/h

Subset interval: X2= < flow 2.25 54 > m3/h

3.3.1.6 Labelled Interval Calculus in Designing for Reliability

An approach to designing for reliability that 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 been 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

Multi-objective optimisation of conceptual 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 different intervals of per-formance values Considering the perper-formance limits represented diagrammatically

in Figs 3.23, 3.24 and 3.25, where an example of two performance limits, one upper performance 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 Limit 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 i j is the value closest to the nominal design condition that

ap-proaches either minimum or maximum limit interval The value of x i j 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 i j approaching the maximum limit interval is the performance variable

of temperature

x i j=Max Temp T1− Nom T High (×20)

Max Temp T1− Min Temp T2

(3.83)

Given relationship: dataset:

(Max Temp T1− Nom T High)/(Max Temp T1− Min Temp T2) × 20

where

Max Temp T1= maximum performance interval

Min Temp T2 = minimum performance interval

Nom T High = nominal performance interval high

Labelled intervals:

Max Temp T1= < all-parts only T1t1lt1h>

Min Temp T2 = < all-parts only T2t2lt2h>

Nom.T High = < all-parts only THtHltHh>

where

t1l = lowest temperature value in interval of

maximum performance interval

t1h= highest temperature value in interval of

maximum performance interval

t2l = lowest temperature value in interval of

minimum performance interval

t2h= highest temperature value in interval of

minimum performance interval

tHl = lowest temperature value in interval of

nominal performance interval high

tHh = highest temperature value in interval of

nominal performance interval high

Computation: propagation rule 1:

(only X ) and (only Y ) and G ⇒ (only Range (G, X, Y ))

x i j [corners (Max Temp T1, Nom T High, Min Temp T2)]

= (t1h−tHl/t1l−t2h) × 20 , (t1h−tHl/t1l−t2l) × 20 ,

(t1h−tHl/t1h−t2h) × 20 , (t1h−tHl/t1h−t2l) × 20 ,

(t1l−tHl/t1l−t2h) × 20 , (t1l−tHl/t1l−t2l) × 20 ,

(t1l−tHl/t1h−t2h) × 20 , (t1l−tHl/t1h−t2l) × 20 ,

(t1h−tHh/t1l−t2h) × 20 , (t1h−tHh/t1l−t2l) × 20 ,

(t1h−tHh/t1h−t2h) × 20 , (t1h−tHh/t1h−t2l) × 20 ,

(t1l−tHh/t1l−t2h) × 20 , (t1l−tHh/t1l−t2l) × 20 ,

(t1l−tHh/t1h−t2h) × 20 , (t1l−tHh/t1h−t2l) × 20 ,

x i j [range (Max Temp T1, Nom T High, Min Temp T2)]

= (t1l−tHh/t1h−t2l) × 20 , (t1h−tHl/t1l−t2h) × 20

Propagation result:

x i j = < all-parts only

x i j (t1l−tHh/t1h−t2l) × 20 , (t1h−tHl/t1l−t2h)×20 >

where x i jis 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 i j approaching the maximum limit interval, with x i j

in the range (Max Temp T1, Nom T High, Min Temp T2), and the data point x i j

being dimensionless, has a propagation result equivalent to the following labelled interval:

< all-parts only xi j (t1l−tHh/t1h−t2l)×20 , (t1h−tHl/t1l−t2h)×20 > , which

represents the relationship:

x i j=Max Temp T1− Nom T High (×20)

Max Temp T1− Min Temp T2

In the case of the data point x i j approaching the minimum limit interval, where

the performance variable is temperature

x i j=Nom T Low − Min Temp T2(×20)

Max Temp T − Min Temp T (3.84)

Given relationship: dataset:

(Max Temp T1− Nom T High)/(Max Temp T1− Min Temp T2) × 20

where

Max Temp T1= maximum performance interval

Min Temp T2 = minimum performance interval

Nom T Low = nominal performance interval low

Labelled intervals:

Max Temp T1= < all-parts only T1t1lt1h>

Min Temp T2 = < all-parts only T2t2lt2h>

Nom T Low = < all-parts only TLtLltLh>

where

t 1i = lowest temperature value in interval of

maximum performance interval

t1h = highest temperature value in interval of

maximum performance interval

t2l = lowest temperature value in interval of

minimum performance interval

t2h = highest temperature value in interval of

minimum performance interval

tLl = lowest temperature value in interval of

nominal performance interval low

tLh= highest temperature value in interval of

nominal performance interval low

Computation: propagation rule 1:

(only X ) and (only Y ) and G ⇒ (only Range (G, X, Y ))

x i j [corners (Max Temp T1, Nom T High, Min Temp T2)]

= (tLh−t2l/t1l−t2h) × 20 , (tLh−t2l/t1l−t2l) × 20 ,

(tLh−t2l/t1h−t2h) × 20 , (tLh−t2l/t1h−t2l) × 20 ,

(tLl−t2l/t1l−t2h) × 20 , (tLl−t2l/t1l−t2l) × 20 ,

(tLl−t2l/t1h−t2h) × 20 , (tLl−t2l/t1h−t2l) × 20 ,

(tLh−t2h/t1l−t2h) × 20 , (tLh−t2h/t1l−t2l) × 20 ,

(tLh−t2h/t1h−t2h) × 20 , (tLh−t2h/t1h−t2l) × 20 ,

(tLl−t2h/t1l−t2h) × 20 , (tLl−t2h/t1l−t2l) × 20 ,

(tLl−t2h/t1h−t2h) × 20 , (tLl−t2h/t1h−t2l) × 20 ,

x i j [range (Max Temp T1, Nom.T High, Min Temp T2)]

= (tLl−t2h/t1h−t2l) × 20 , (tLh−t2l/t1l−t2h) × 20

Propagation result:

x i j = < all-parts only

x i j (tLl−t2h/t1h−t2l) × 20 , (tLh−t2l/t1l−t2h) × 20 >

where x i jis 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 i j approaching the minimum

limit interval, with x i j in the range (Max Temp T1, Nom T High, Min Temp.

T2), and x i j dimensionless, has a propagation result equivalent to the following labelled interval:

< all-parts only xi j (tLl−t2h/t1h−t2l) × 20 , (tLh−t2l/t1l−t2h) × 20 >

which represents the relationship:

x i j=Nom T Low − Min Temp T2(×20)

Max Temp T1− Min Temp T2

b) Determination of a Data Point: One Upper Limit Interval

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 i japproaching the upper limit interval

x i j=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 s1ls1h>

LSI = lowest stress interval < all-parts only LSI s2ls2h>

NSI= nominal stress interval < all-parts only NSI sHlsHh>

where:

s1l = lowest stress value in interval of highest stress interval

s1h = highest stress value in interval of highest stress interval

s2l = lowest stress value in interval of lowest stress interval

s2h = highest stress value in interval of lowest stress interval

sHl = lowest stress value in interval of nominal stress interval

sHh = highest stress value in interval of nominal stress interval

Computation: propagation rule 1:

(only X ) and (only Y ) and G ⇒(only Range (G, X, Y))

x i j[corners (HSL, NSL, LSL)]

= (s1h− sHl/s1l− s2h) × 10 , (s1h− sHl/s1l− s2l) × 10 ,

(s1h− sHl/s1h− s2h) × 10 , (s1h− sHl/s1h− s2l) × 10 ,

(s1l− sHl/s1l− s2h) × 10 , (s1l− sHl/s1l− s2l) × 10 ,

(s1l− sHl/s1h− s2h) × 10 , (s1l− sHl/s1h− s2l) × 10 ,

(s1h− sHh/s1l− s2h) × 10 , (s1h− sHh/s1l− s2l) × 10 ,

(s1h− sHh/s1h− s2h) × 10 , (s1h− sHh/s1h− s2l) × 10 ,

(s1l− sHh/s1l− s2h) × 10 , (s1l− sHh/s1l− s2l) × 10 ,

(s1l− sHh/s1h− s2h) × 10 , (s1l− sHh/s1h− s2l) × 10 ,

x i j[range (HSL, NSL, LSL)]

= (s1l− sHh/s1h− s2l) × 10 , (s1h− sHl/s1l− s2h) × 10

Propagation result:

x i j = < all-parts only

x i j (s1l− sHh/s1h− s2l) × 10 , (s1h− sHl/s1l− s2h) × 10 >

where x i jis dimensionless

Description:

The data point x i j approaching the upper limit interval, with x i jin the range (High

Stress Level, Nominal Stress Level, Lowest Stress Level), and x i jdimensionless, has a propagation result equivalent to the following labelled interval:

< all-parts only xi j (sLl− s2h/s1h− s2l) × 20 , (sLh− s2l/s1l− s2h) × 20 > ,

which represents the relationship:

x i j=Highest Stress Level− Nominal Stress Level (×10)

Highest Stress Level− Lowest Stress Est.

c) Determination of a Data Point: One Lower Limit Interval

In the case of the data point x i japproaching the lower limit interval

x i j=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 C1 = maximum capacity interval

Min Cap C2 = minimum capacity interval

Nom Cap C = nominal capacity interval low

Labelled intervals:

Max Cap C1 = < all-parts only C1c1lc1h>

Min Cap C2 = < all-parts only C2c2lc2h>

Nom Cap CL= < all-parts only C L cLlcLh>

where

c1l = lowest capacity value in interval of maximum

capacity interval

c1h = highest capacity value in interval of maximum

capacity interval

c2l = lowest capacity value in interval of minimum

capacity interval

c2h = highest capacity value in interval of minimum

capacity interval

cLl = lowest capacity value in interval of nominal capacity

interval low

cLh = highest capacity value in interval of nominal

capacity interval low

Computation: propagation rule 1:

(only X ) and (only Y ) and G ⇒ (only Range (G, X, Y ))

x i j [corners (Max Cap Min Cap C2, Nom Cap C L)]

= (cLh− c2l/c1l− c2h) × 10 , (cLh− c2l/c1l− c2l) × 10 ,

(cLh− c2l/c1h− c2h) × 10 , (cLh− c2l/c1h− c2l) × 10 ,

(cLl− c2l/c1l− c2h) × 10 , (cLl− c2l/c1l− c2l) × 10 ,

(cLl− c2l/c1h− c2h) × 10 , (cLl− c2l/c1h− c2l) × 10 ,

(cLh− c2h/c1l− c2h) × 10 , (cLh− c2h/c1l− c2l) × 10 ,

(cLh− c2h/c1h− c2h) × 10 , (cLh− c2h/c1h− c2l) × 10 ,

(cLl− c2h/c1l− c2h) × 10 , (cLl− c2h/c1l− c2l) × 10 ,

(cLl− c2h/c1h− c2h) × 10 , (cLl− c2h/c1h− c2l) × 10 ,

x i j [range (Max Cap Min Cap C2, Nom Cap C L)]

= (cLl− c2h/c1h− c2l) × 10 , (cLh− c2l/c1l− c2h) × 10

Propagation result:

x i j = < all-parts only

x i j (cLl− c2h/c1h− c2l) × 10 , (cLh− c2l/c1l− c2h) × 10 >

where x i jis 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:

The data point x i j approaching the lower limit interval, with x i jin the range (Max.

Capacity Level, Min Capacity Level, Nom Capacity Level), and x i j dimension-less, has a propagation result equivalent to the following labelled interval:

< all-parts only xi j (cLl− c2h/c1h− c2l) × 10 , (cLh− c2l/c1l− c2h) × 10 > with x i j in the range (Max Cap Min Cap C2, Nom Cap C L), representing the relationship:

x i j=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 process 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 i j refers to a single parameter Similarly, in the expanded method using labelled interval calculus (LIC), the performance measures of each system of

a process are described in an interval matrix form, containing datasets relating 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 of performance values

A parameter performance index, PPI, can be calculated for each row

PPI= n



n

∑

j=1

1

x i j

−1

(3.87)

where n is the 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 i j

−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

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 intervals in each column (i.e the number of performance

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 i j = < all-parts only

x i j (x1l− xHh/x1h− x2l) × 10 , (x1h− xHl/x1l− x2h) × 10 >

Table 3.10 Labelled intervals for specific performance parameters

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) < 1.1 8.3 > < 1.3 6.7 > < 1.3 6.7 > < 1.1 8.3 >

Pressure (kPa) < 2.2 8.8 > < 2.2 6.9 > < 2.2 6.9 > < 1.9 7.5 >

Temp (◦C) < 2.0 10.0 > < 1.7 7.5 > < 1.7 7.5 > < 1.7 5.0 >

Labelled intervals—flow:

Vessel interval: = < all-parts only x i j1.1 8.3 >

Pump 1 interval:= < all-parts only x i j1.3 6.7 >

Pump 2 interval:= < all-parts only x i j1.3 6.7 >

Pump 3 interval:= < all-parts only x i j1.1 8.3 >

Labelled intervals—pressure:

Vessel interval: = < all-parts only x i j2.2 8.8 >

Pump 1 interval:= < all-parts only x i j2.2 6.9 >

Pump 2 interval:= < all-parts only x i j2.2 6.9 >

Pump 3 interval:= < all-parts only x i j1.9 7.5 >

Labelled intervals—temperature:

Vessel interval: = < all-parts only x i j2.0 10.0 >

Pump 1 interval:= < all-parts only x i j1.7 7.5 >

Pump 2 interval:= < all-parts only x i j1.7 7.5 >

Pump 3 interval:= < all-parts only x i j1.7 5.0 >

The parameter performance index, PPI, can be calculated for each row

PPI= n



n

∑

j=1

1

x i j

−1

(3.90)

where n is the number of systems in row i.

Labelled intervals:

Flow (m3/h) PPI = < all-parts only PPI 1.2 7.4 >

Pressure (kPa) PPI= < all-parts only PPI 2.1 7.5 >

Temp (◦C) PPI = < all-parts only PPI 1.8 7.1 >

The system performance index, SPI, can be calculated for each column

SPI= m



m

∑

i=1

1

x i j

−1

(3.91)

where m is the number of parameters in column i.

Labelled intervals:

Vessel SPI = < all-parts only 1.6 9.0 >

Pump 1 SPI= < all-parts only 1.7 7.0 >

Pump 2 SPI= < all-parts only 1.7 7.0 >

Pump 3 SPI= < all-parts only 1.5 6.6 >

Description:

The parameter performance index, PPI, and the system 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)