The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 1

ISSN 2229-5518

Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults

Rajeev Kumar and Pooja Bhatia

ABSTRACT—The paper deals with a model developed for a single centrifuge system working in Thermal Power Plant, Panipat (Haryana) India, which has alternate periods of operation and rest. The system may have minor, neglected and major faults. It is assumed that the occurrence of a minor fault leads to degradation of the system whereas occ urrence of a major fault leads to failure of the system. The neglected faults that are in the system are generally neglected for repair during operation of the system until the system goes to rest or complete failure and the system has to be stopped on occurrence of minor fault for repair. Various measures of system effectiveness are obtained regarding the reliability and cost analysis of the system is carried out and the conclusions on the basis of the graphical studies are given.

KEY W ORDS—centrifuge system, mean time to system failure, neglected faults, Markov process, profit, regenerative point technique.

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1 INTRODUCTION

N the field of reliability modeling, several different types of systems considering various aspects such as types of failure (faults) repairs, inspection polices, modes of operations, switching etc. have been analyzed by several researchers

including [1],[2],[3],[4],[5],[6], [7].

In many practical situations, for instance in thermal

power plant for oil purification, milk plants for making butter,

laboratories, blood fractionation, wine clarification, etc.

centrifuge systems are used and act as the main systems or

sub-systems. In these situations the reliability and cost of

centrifuge systems play a very important and crucial role.

It was observed, while collecting real data on faults/

failures and repairs on a centrifuge system working in Thermal

Power plant, Panipat (Haryana) which undergoes periodic rest

(normally after eight hours), that a minor fault leads to

degradation of the system whereas a major fault leads to

complete failure of the system. Some faults such as vibration,

abnormal sound, etc are generally neglected for repair during

the operation of the system until system goes to rest or to

complete failure. Sometimes these neglected faults also lead to

complete failure of the system. Further the system has to be

stopped on occurrence of minor fault. The cost and

maintenance analysis of centrifuge system considering the

aspects of periodic rest period, neglected faults and stoppage

on occurrence of minor faults has not been reported in the

literature of reliability so far. However, the reliability and

availability analyses of a centrifuge system considering minor,

ignored and major faults has been carried out by [8], [9].

Keeping above in view, the present paper deals with a

single unit centrifuge system considering major, minor and

neglected faults wherein a minor fault degrades the system

whereas a major fault leads to complete failure of the system. The neglected fault is taken as the fault that may be neglected for repair during the operation of the system until system goes to rest or to complete failure. The system undergoes periodic rest. During the rest period or complete failure, the repairman first inspect whether the fault is repairable or non repairable and accordingly carry out repair or replacement of the faulty components. Sometimes when minor fault occurs, the system has to be stopped. Various measures of system effectiveness, such as mean sojourn times, mean time to system failure, expected uptime, busy period of repairman and expected profit are obtained using Markov processes and regenerative point technique. Various conclusions regarding the reliability and cost of the system on the basis of graphical analyses are drawn.

negligible time, whenever called for repair.

replacement.

and restart of the system are exponential whereas

other time distributions are general.

λ1/λ2/λ3 Rate of occurrence of major/minor/ neglected faults

a/b Probability that the fault is non repairable /

repairable, b = 1- a

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The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 2

ISSN 2229-5518

p/q Probability that the neglected fault lead to/

don’t lead to complete failure, q=1-p

1 Rate at which the unit goes to rest

2 Rate at which the unit is stopped

β Rate at which the unit restarts after rest

i1(t)/i2(t)/i3(t) p.d.f. of time to inspection of the unit at

failed/stopped/rest state

I1(t)/I2(t)/I3(t) c.d.f. of time to inspection of the unit at

failed/stopped/rest state

g1(t)/g2(t)/g3(t) p.d.f. of time to repair the unit at failed /

stopped/rest state

Operative State Failed State Degraded State

Rest State Stopped State

The transition probabilities are

G1(t)/G2(t)/G3(t) c.d.f. of time to repair the unit at failed /

stopped/rest state

dQ01 (t)

e ( 1 2 3 1 ) t dt

( 1 2 3 1 ) t

dQ02 (t)

e ( 1 2 3 1 ) t dt

( 1 2 3 1 ) t

h1(t)/h2(t)/h3(t) p.d.f. of time to replacement of the unit at failed/stopped/rest state

dQ03 (t)

3e dt

dQ04 (t) e dt

H1(t)/H2(t)/H3(t) c.d.f. of time to replacement of the unit at

dQ15 (t) ai1 (t)dt

2 ( t )

dQ16 (t) bi1 (t)dt

failed/stopped/rest state

dQ27 (t)

2 e dt

dQ31 (t) pk(t)dt

( t )

k(t)/K(t) p.d.f./c.d.f. of time to delay in repair of the

neglected fault

O/R Operative / Rest state

Or/On/ Od Operative state under inspection/ neglected fault/degradation

Ri /Rr/Rrp Rest state under inspection/repair/

replacement

dQ34 (t) qk(t)dt

dQ (t) ai (t)e- ( t ) dt

dQ50 (t) h1 (t)dt dQ78 (t) ai3 (t)dt dQ80 (t) h3 (t)dt dQ10,4 (t) h2 (t)dt

dQ40 (t) e I2 (t)dt

dQ (t) bi (t)e ( t ) dt

dQ60 (t) g2 (t)dt dQ79 (t) bi3 (t)dt dQ90 (t) g1 (t)dt dQ11, 4 (t) g3 (t)dt

lim Q** (s)

Fi/ Fr / Frp Failed state under inspection/ repair/

The non-zero elements pij are

pij

s 0 ij

replacement

A diagram showing the various states of transition of the system is shown in Fig. 1. The epochs of entry in to state 0, 1, 2,

p01

p03

1

1 2 3 1

3

1 2 3 1

p02

p04

2

1 2 3 1

1

1 2 3 1

3, 4, 5, 6, 7, 8, 9, 10, 11 are regenerative point and thus all the states are regenerative states.

p15

ai* (0)

*

p bi* (0)

*

p31

p40 p4,11 p60 p79 p90

p11, 4

pk (0)

1 i * ( ) ai * ( ) g* (0) bi* (0) g* (0)

g * (0)

p34

p4,10 p50 p78 p80 p10, 4

qk (0)

bi * ( )

h * (0) ai* (0) h * (0) h * (0)

By these transition probabilities, it can be verified that p01 + p02 + p03 + p04 = p15 + p16 = p34 + p31 = p4,10 +

p4,11 + p40 = p78 + p79 = 1, p27 = p50 = p60 = p80 = p90

= p10,4 = p11,4 = 1

The mean sojourn time in the regenerative state i(µi) is defined as the time of stay in that state before transition to any other state then we have

1

0 =

1 2 3 1

* /

1 = 1

1

(0) 2=

2

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The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 3

ISSN 2229-5518

*/ *

* / For graphical analysis the following particular cases are

3 = k (0) 4 = 1 i2 ( ) 5=

h1 (0)

considered:

= g*/ (0) µ =

i* / (0) =

* (0)

g1 (t)

1 ( t )

1

g2 (t)

2 ( t )

2

g3 (t)

3 ( t )

3

= g*/ (0) =

* (0) 11 =

* (0)

k(t) e

( t )

3 ( t )

h1 (t) 1e

1 ( t )

( t )

h2 (t) 2 e

2 ( t )

( t )

The unconditional mean time taken by the system to transit for any regenerative state j, when it is counted from

h3 (t)

i3 (t)

3e

3 ( t )

3

i1 (t) 1e

i2 (t) 2 e

epoch of entrance into that state i, is mathematically stated as- mij = td Qij(t)

Various graphs are drawn for the MTSF and the profit (P)

for the different values of the rates of occurrence of major,

Thus,

m01 + m02 + m03 + m04 = 0 m15 + m16 = 1

m27 = 2 m34 + m31 = 3

m40 + m4,10 + m4,11 = 4 m50 = 5

m60 = 6 m78 + m79 = 7

m80 = 8 m90 = 9

m10,4 = µ10 m11,4 = µ11

Using probabilistic arguments for regenerative processes, various recursive relations are obtained and are solved to derive important measures of the system effectiveness that are as given below:

where

N1 = µ0 + µ2p02 + µ3p03

N2= p40 [p02p27 µ7+ (p01+p03p31) µ1 + (p03p34+p04) µ 4]

N3= p40[p02µ2+p16 µ6(p01+p03p31)+p02p27p79 µ9]

+ p4,11µ11(p03p34+p04)

N4= p40[p02p27p78 µ8+ p15µ5(p01+p03p31)]+ µ10p4,10(p03p34+p04)

D1= p40 [µ0 + p03 µ3 + p02 (µ2 + p79µ9 + p78µ8 + µ7) +

(p01+p03p31) (µ1+p15 µ5+ p16µ6)] +

(µ11p4,11+ µ10p4,10 + µ4) (1-p01-p02- p03p31)

The expected profit incurred of the system is

P = C0 A0 C1 Bi C2 Br C3Brp-C

where

C0 = revenue per unit uptime of the system

C1 = cost per unit time of inspection

C2 = cost per unit time of repair

C3 = cost per unit time of replacement

C = cost of installation of the unit

minor and neglected faults (λ1, λ2, λ3), repair rates (β1, β2, β3), replacement rates (γ1, γ2, γ3), inspection rates (α1, α2, α3) and

rates of rest, stoppage and delay for repair ( 1, 2,δ) of the unit.

Fig. 2 and Fig.3 give the graphs between MTSF (To) and

the rate of occurrence of major and neglected faults (λ1 & λ3), respectively for different values of rate of occurrence of minor faults (λ2) and rate at which the system has to be stopped (η2). The graph reveals that the MTSF deceases with increase in the values of rates of occurrence of major, minor and neglected faults, respectively. Further it may also be concluded from the Fig.3 that the delay in the repair of neglected faults result into decrease in the values of MTSF.

Fig. 2

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The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 4

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Fig.3

The graph in Fig. 4 shows the pattern of profit with respect to rates of occurrence of minor faults for different values of major faults (λ2 & λ1). The curves in the graph indicate that the profit of the system decreases with the increase in the values of the rates of occurrence of minor faults for different values of major faults.

Fig.4

The curves in the Fig.5 show the behavior of the profit with respect to rate of occurrence of neglected faults (λ3) of the system for the different values of rate of delay in repair of neglected faults (δ). It is evident from the graph that profit decreases with the increase in the rate due to occurrence of neglected faults. From the Fig. 5 it may also be observed that for δ =2, the profit is > or = or < 0 according as λ3 is < or = or

>.879. Hence the system is profitable to the company whenever λ3 ≤ .879. Similarly, for δ =3 and δ =4 respectively the profit is > or = or < 0 according as λ3 is < or = or > .825 and .8 respectively. Thus, in these cases, the system is profitable to the company whenever λ3 ≤ .825 and .8 respectively.

Fig. 5

The graph in Fig. 6 shows the pattern of profit with respect to the rates of occurrence of neglected faults for different values of rate of system has to be stopped (λ3,η2). The curves in the graph indicate that the profit of the system decreases with the increase in the values of the rate of occurrence of neglected faults as well as rate with which system is stopped.

Fig. 6

The curves in the Fig.7 show the behavior of the profit with respect to the revenue per unit up time (CO) of the system for the different values of rate of occurrence of neglected faults (λ3) due to neglected faults. It is evident from the graph that profit increases with the increase in revenue up time of the system for fixed value of the rate of occurrence of neglected faults. From the Fig.7 it may also be observed that for λ3 =

0.001, the profit is > or = or < 0 according as CO is > or = or < Rs.20664.21. Hence the system is profitable to the company whenever CO ≥ Rs. 20664.21. Similarly, for λ3 = 0.201 and λ3 =

0.401 respectively the profit is > or = or < 0 according as CO is > or = or < Rs.24093.56 and Rs.28257.06 respectively. Thus, in these cases, the system is profitable to the company whenever CO ≥ Rs. 24093.56 and Rs. 28257.06 respectively.

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The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 5

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Fig. 7

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[6] Kumar, R. and Bhatia, P., “Reliability and availability analysis of a single unit centrifuge system considering various faults,” International conference on science and engineering (ICSE11), pp.536-539 2011.

[7] Murari, K. and Goyal, V.,” Reliability system with two types of repair facilities,” *Microelectronics Reliability*, 23(6), pp.1015-1025 1983.

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[9] Taneja, G., Tayagi, V. K. and Bhardwaj, P.,” Profit analysis of a single unit programmable logic controller,” *Pure and Applied Mathematical Sciences*, LX(1-2), pp.55-71,2004.

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