Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 50 ppt

Table 4.6 Remaining capacity versus unavailable subgroupsSub-system Number of Subgroup Remaining capacity as n subgroups become unavailable group subgroups capacity n= 0 n= 1 n= 2 n= 3 T

Table 4.5 Process capacities per subgroup

Sub-system group No of subgroups Capacity per subgroup

• Process limitations

• Quality limitations, etc.

Each state in the simple power plant example has only one subgroup that is the limiting factor, or bottleneck, for the plant’s power output capability in that state This constraint is illustrated in Fig 4.35 where the example plant is represented

as a set of pipes and valves of varying capacities Each section of pipe and valve corresponds to a subgroup in which the subgroup’s unavailability is analogous to

a valve being closed:

• The single A subgroup (consisting of two coal bin sub-systems) is wide enough

to handle 100% of the flow;

• Each of the two B subgroups (consisting of the two slurry mill sub-systems) is

wide enough to handle 75% of the flow;

• Each of the three C subgroups (consisting of three gas turbines and three

gener-ators) is wide enough to handle 50% of the flow

For example, if two C subgroups are unavailable, and one B subgroup is un-available, the C subgroup is the limiting factor because its remaining capacity is only 50%, whereas the remaining capacity in any one of the B subgroups is 75%

Furthermore, when two C subgroups are unavailable, there could be either no un-available B subgroups or one unun-available B subgroup, without further reducing the

process flow from the resulting 50% output brought about by the one available C subgroup

f) Defining Different States

(1) Table 4.5 shows the percentage of the plant’s process flow capability that each type of subgroup could support

(2) Table 4.6 shows the reduction in plant flow capacity as the number of unavail-able subgroups in each sub-system group increases, given that all other subgroups are available Where excess capacity beyond 100% exists in a subgroup, 100% is given as the throughput capacity

(3) Table 4.7 shows the flow capacities and state definitions The flow capacities are taken from the previous table Note that, although the 100% entry appears four times, it is entered only once in the table below All flow capacities other than 100% are entered as many times as they appear in the previous table Thus, the 0% flow capacity is entered three times The capacities should be entered in decreasing order

Table 4.6 Remaining capacity versus unavailable subgroups

Sub-system Number of Subgroup Remaining capacity as n subgroups become unavailable

group subgroups capacity n= 0 n= 1 n= 2 n= 3

Table 4.7 Flow capacities and state definitions of unavailable subgroups

State number Flow capacity Unavailable subgroups

to simplify the state definition process The entries under columns A, B and C in

Table 4.7 must be the same as the entries under columns n = 0, n = 1, n = 2 and n = 3

in Table 4.6

Process of entering the different state definitions

i) Enter for each sub-system group the number of unavailable subgroups that would still allow 100% process flow In the example, no unavailable subgroups

in sub-system group A would allow for 100% process flow Similarly, no sin-gle subgroup in sub-system group B would allow for 100% process flow In sub-system group C, either zero or one unavailable subgroup allows for 100% process flow

ii) Enter for each state the number of unavailable subgroups in the appropriate sub-system group that are responsible for that state’s capacity For example, the 75% capacity of state 2 is the result of one of system group B’s sub-groups being unavailable; the 50% capacity of state 3 is the result of two

of sub-system group C’s subgroups being unavailable; the 0% capacity of states 4, 5 and 6 each is the respective result that one of A’s subgroups, or two of B’s subgroups, or three of C’s subgroups are unavailable This is indi-cated in Table 4.8

iii) For each state that has a non-zero flow capacity, enter the subgroups in each re-maining sub-system group that could be unavailable without further decreas-ing the flow capacity of that state For example, state 3 has a 50% flow ca-pacity because of unavailability of two of C’s subgroups Zero subgroups of sub-system group A can be unavailable, and either zero or one of sub-system group B’s subgroups can be unavailable without decreasing the flow capacity

of 50% for state 3

Table 4.8 Flow capacities of unavailable sub-systems per sub-system group

State number Flow capacity Unavailable subgroups

Table 4.9 Unavailable sub-systems and flow capacities per sub-system group

State number Flow capacity Unavailable subgroups

Table 4.10 Unavailable sub-systems and flow capacities per sub-system group: final summary

State number Flow capacity Unavailable subgroups

iv) The remaining entries to be made are in the 0% capacity states These remain-ing entries indicate the number of subgroups that can be unavailable in each sub-system group in conjunction with other sub-system groups, where a 0% capacity state can be defined This is indicated in Table 4.9 The final summary

is indicated in Table 4.10

g) Evaluating Complexity of the Different State Definitions

One of the more significant challenges of engineering design is to provide a rational account of the uncertainty surrounding the state events of unavailable systems that could be responsible for diminishing a design’s capacity and/or performance Classi-cal probability theory offers a feasible approach but it is burdened with well-known

epistemological flaws, considered in Sect 3.3.2 (Zadeh 1995; Laviolette et al 1995).

Theories of fuzzy sets and possibility represent attempts to rectify some of the

de-ficiencies in classical probability theory (Dubois et al 1993) However, all of these

theories fundamentally accept the basic fact that random variables form a significant

part of uncertainty

Consider the state events of unavailable systems that diminish the overall

ca-pacity of the example power plant: Let x i represent the sub-system states listed in

Table 4.10 where i = states 1,2,3, ,6 Furthermore, let yθj represent the state events of unavailable sub-system groups that could affect the overall capacity of

the example power plant, where the subscriptsθ =sub-system group A, B or C,

and j = subgroup 1, 2, 3 Individual elements of x i can then be combined into

a primary set of state events of unavailable sub-systems, denoted by X , and the elements yθj can be combined into a secondary set of state events denoted by Y

A graphical representation of these elements is called a complex, whereby each x i

element is taken as the vertex of a surface formed by connected points representing the possible state events of each related subgroup yθj , the state event elements of Y , which are called simplices (Casti 1994).

Thus, the system states represented by x iare:

X = {x1,x2,x3,x4,x5,x6} (4.181)

and the possible state events represented by yθjare:

Y = {y A0 ,y A1 ,y B0 ,y B1 ,y B2 ,y C0 ,y C1 ,y C2 ,y C3 } (4.182)

The outcomes of the compound events resulting from the integration of systems

forming each subgroup (as depicted in the availability block diagram of Fig 4.27)

are given by the values (expressed as percentages of the overall capacity of the example power plant) of the system states represented by x i , and are called random variables According to Table 4.10, outcomes of the compound events are:

x1= (y A0 + y B0 + y C0 ,y C1 ,y C2 ,y C3)

= 100%

x2= (y B1 ,y B2 ,y B1 ,+y C1 ,y B1 + y C2 ,y B1 + y C3 ,

y B2 + y C1 ,y B2 + y C2 ,y B2 + y C3)

= 75%

x3= (y B1 + y C1 + y C2 ,y B1 + y C1 + y C3 ,y B1 + y C2 + y C3 ,

y C1 + y C2 ,y C1 + y C3 ,y C2 + y C3)

= 50%

x4= (y A1)

= 0%

Table 4.11 Unavailable subgroups and flow capacities incidence matrix

State number Flow capacity Unavailable subgroups

1 100% (y A0 + y B0 + y C0 ,y C1 ,y C2 ,y C3)

2 75% (y B1 ,y B2 ,y B1 ,+y C1 ,y B1 + y C2 ,y B1 + y C3 ,y B2 + y C1 ,

y B2 + y C2 ,y B2 + y C3)

3 50% (y B1 + y C1 + y C2 ,y B1 + y C1 + y C3 ,y B1 + y C2 + y C3 ,y C1 + y C2 ,

y C1 + y C3 ,y C2 + y C3)

Table 4.12 Probability of incidence of unavailable systems and flow capacities

State number Flow capacity Unavailable subgroups Probability of incidence

x5= (y B1 + y B2)

= 0%

x6= (y C1 + y C2 + y C3)

= 0%

Taking the elements of X to be the vertices of the unavailability complex of the power plant, and denoting the elements of Y to be simplices formed from these vertices, the relation R Y linking the two sets can be established, such that the pairs

of elements(yθj ,x i ) are in the relation R Y if, and only if, the possible state events

of unavailable subgroups, yθj , form part of the elementary system states x i Thus,

(y C1 ,x1) is in R Y; however,(y A1 ,x1) and (y B1 ,x1) are not

Computing all the chains of connections in this complex enables the formation

of an incidence matrix for R Y This matrix is the kind of incidence structure for which classical probability theory works well to express the concept of uncertainty

in evaluating the integrity of the design

The complex of which the simplices are the state event elements of Y represents the sample space of the various unavailability states, expressed as percentages of

the overall capacities, as indicated in Table 4.11 The probability of system unavail-ability incidence is given in Table 4.12

h) Evaluation of Alternatives

At this point in systems engineering analysis, alternative design solutions that sat-isfy system constraints are developed Effectiveness measures are initially quantified for each solution without serious consideration of cost Later, both effectiveness and costs are evaluated After alternative system configurations have been synthesised and the effectiveness requirements have been established for each alternative, they can be compared A typical trade-off matrix technique is appropriate In most stud-ies, the analysis is restricted to an evaluation of cost and to some physical attributes

of the system such as reliability, availability, maintainability or safety It is, how-ever, necessary to analyse cost and effectiveness in monetary terms An adequate analysis cannot be performed unless both parts of the relationship are evaluated in commensurate terms—i.e when evaluating on the basis of costs, all comparisons must be kept in terms of costs Prior to such a cost versus effectiveness compari-son, however, it is necessary to determine the physical attributes of the system (i.e

system integrity).

The following example indicates how overall system integrity can be determined through systems engineering analysis to obtain the system’s sub-system and/or as-sembly attributes of mean times between failures and failure repair times

Figure 4.36 represents a process block diagram (i.e a simplified process flow diagram) of a turbine/generator system

After the development of an availability block diagram (ABD), the overall in-tegrity of the system can be determined based on the ABD configuration and at-tributes of the system’s sub-systems and/or assemblies (Table 4.13)

An ABD of the super-heated steam turbine/generator system illustrated in the process block diagram of Fig 4.36 is given in Fig 4.37

Determining overall mean time to repair (MTTR system) From the integrity

values given in Table 4.13:

MTTR system=Σ(λR)

where: λ= failure rate

R = repair time (h)

MTTR system= 39,227/370.43

MTTR system= 105.9

Determining overall mean time between failures (MTBF system) From the

in-tegrity values given in Table 4.13:

MTBF system = 2.699

Unit 1

boiler

Feed water

heater

Boiler

feed

pump

Hot well pump

Cond.

pump

Unit 1 turbine

Condenser

De-aerator

Super heater

Unit 1 generator

Fig 4.36 Process block diagram of a turbine/generator system

Power generating system

A = 96.2 MTBF = 2.699 MTTR = 105.9

MTBF = 8.965

R = 124.5

MTBF = 20.653

R = 142.5

MTBF = 51.046

R = 148.6

MTBF = 62.344

R = 39.5

MTBF = 36.063

R = 48.3

MTBF = 85.616

R = 42.5

MTBF = 91.408

R = 96.3

MTBF = 112.306

R = 98.5

MTBF = 8.652

R = 96.3

Steam condenser

Boiler feed pump

Boiler

Fig 4.37 Availability block diagram of a turbine/generator system, where A= availability, MTBF = mean time between failure (h), MTTR = mean time to repair (h)

Table 4.13 Sub-system/assembly integrity values of a turbine/generator system

Power system Failure rate MTBF Repair rate λ× R

items ( λ fail/106 h) (106/λ h) (R ,h)

1 Generator 111.55 8.965 124.5 13.888

2 Turbine 48.42 20.653 142.5 6.900

3 Hot pump 27.73 36.062 48.3 1.339

4 Condenser 19.59 51.046 148.6 2.911

5 Cond pump 16.04 62.344 39.5 0.634

6 De-aerator 10.94 91.408 96.3 1.053

7 Feed pump 11.68 85.616 42.5 0.496

8 Feed heater 8.90 112.306 98.5 0.876

Determining overall availability (A system) From the integrity values given in

Table 4.13, and from the formula for steady-state availability, we get:

2.699 + 105.9

A= 96.2%

where, in Eqs (4.184) and (4.185):

λ = failure rate

A = availability

MTBF= mean time between failure (h)

MTTR= mean time to repair (h)

4.3.3.4 Evaluating Complexity in Engineering Design

With the phenomenal advancement in process technology, there has been an almost similar increase in the complexity of engineered installations, particularly large in-tegrated systems Much engineering effort has gone into analysing and

understand-ing systems complexity in an attempt to try and manage or reduce it at the design

stage Relatively recent research has shown, however, that the real issue is not so

much reducing systems complexity but, rather, reducing complicatedness This is

an important distinction because complexity can, in fact, be a desirable property of

integrated systems, provided it is specifically engineered complexity that reduces complicatedness (Tang et al 2001).

Complexity and complicatedness are not synonymous Complexity is an inher-ent property of systems and the integration of systems; complicatedness is a derived

function of complexity, introduced in the notion of complicatedness of complex sys-tems Equations for each can be developed showing that they are separate and

dis-tinct properties that not only reflect the fundamental behaviour of complex systems but that also provide a design methodology whereby complicatedness can be evalu-ated The implications for systems design engineers are enormous, especially con-cerning complex systems analysis in engineering design The difference between complexity and complicatedness can be illustrated by the following example (Tang

et al 2001)

Relative to a manual transmission, a motor vehicle’s automatic transmission has

more parts and more intricate linkages, making it more complex To the vehicle driver (operator), it is unquestionably less complicated but to the mechanic (main-tainer), who has to repair it, it is more complicated This illustrates a fundamental fact about systems: operational control has an important role on systems to manage their behaviour Complexity, therefore, is an inherent property of systems Compli-catedness is a derived property that characterises the ability to control a complex system A system of complexity level Ca may present different degrees of

compli-catedness K to distinct control units E and F, where:

KE= KE(Ca)

and:

KE,KF= complicatedness of systems E and F

a) Complexity in Systems

There is hardly any research on complicatedness and complexity as distinct prop-erties of systems The focus is on modularisation and integrated interactions with

a bias to linear systems and qualitative metrics Overwhelmingly, the literature

considers systems with a large number of elements as complex (Suh 1999) Very few studies address integrated linkages among the elements (Warfield 2000), and

at least one considers their bandwidth (Tang et al 2001) All these factors are in-herent characteristics of systems; the number of elements, the number of interac-tions among these, and the bandwidth of the interacinterac-tions determine the

complex-ity of the system As these increase, system complexcomplex-ity is expected to increase

For example, consider the system N = {n i } i = 1,2, , p with binary interactions among the elements Complexity C N of this system does not exceed p2, which is denoted by:

C N = Op2

Thus, the system M = {m j } j = 1,2, , p can have complexity:

Thus, when M has {m j × m r } jr and {m j × m r × m s } jrs interactions, then

C M = O(p3)

Furthermore, when M has {m j × m r × m s × m t } jrst interactions, then

C M = O(p4)

This characterisation of complex systems considers systems with feedback loops

of arbitrary nesting (i.e arbitrary loops within loops), and high bandwidth (i.e

vol-ume or number) of interactions among system elements Complexity is a monoton-ically increasing function, as the size of the system and the number of interactions,

as well as the bandwidth of interactions increase (Tang et al 2001)

In the limit, complexity→∞ Complexity is thus defined by:

C = X n∑

b

where:

X is an integer denoting the number of elements {x e }e = 1,2, , p

n is the integer indicated in the relation O (p n)

and:

B1=∑

i j

B2=∑

k

where:

λi j = the number of linkages between x i and x j

βi j = the bandwidth of linkages between x i and x j

λk i j = the number of linkages between x kand(x i ,x j)

βk i j = the bandwidth of linkages between x kand(x i ,x j)

In general:

B n=∑

n

λp i jk n−1βn i jk n−1 (4.191)

where:

λp i jk n−1 is the number of linkages among x

k and (x i ,x j ),(x i ,x j ,x k ), , (x i ,x j ,x k , ,x n −1)

βn i jk n−1 is the bandwidth of linkages for x

k and (x i ,x j ),(x i ,x j ,x k ), , (x i ,x j ,x k , ,x n −1)

B n is a measure of the capacity among the n elements of the system.