CC BY
30
Performance Analysis of System where Service Type for Boiler Depends Upon Major or Minor Failures
Upasana Sharma1 and Rajveer Kaur2* •
department of Statistics, Punjabi University, Patiala-India, [email protected] 2*Department of Statistics, Punjabi University, Patiala-India, Corresponding author: [email protected]
Abstract
In industries, the type of failure sensitively affects the system. So, it is essential to Categorize these failures into different categories to enhance the system performance. In this research, concentration made in differentiating the failure type into major/minor categories with repair/replacement facility for the service by single repairman. Currently, we studied the boiler system of the steam generation plant to perform the task of repair/replacement with a single repairman. A reliability model constructs to compute MTSF(mean time to system failure), availability, Busy period for repair/replacement, and profit evaluation. The above measures were estimated numerically and plotted graphically using semi-Markov processes and regenerative point technique. Various effectiveness measures show how system performance gets affected by major/minor failures & the type of service provided.
Keywords: Regenerative point technique, major/minor failure, repair/replacement, Reliability
modeling, semi-Markov processes.
I. Introduction
Proper functioning of any system is a prerequisite of any industrial process, as an interruption in the operation of a system causes not only deterioration in the quality of manufactured products but damages to the plant itself. Thus the reliability of the system becomes much more essential. Many contributors pay their efforts in the literature of reliability. [5], [8], [2] have worked on cost-effectiveness and reliability analysis on different situations. Various situations on repair, replacement, & inspection have been investigated by the authors [6], [4], [9], [3], [1]. However, the distinction between major and minor faults has not been the topmost research topic in reliability. [7] have discussed the concept of major/minor failures subjected to the ordinary and expert repairman. [10] revealed the possibility of immediate repair on minor failure and waiting time for repair on major failure at night hours. But the concept of repair/replacements depending upon the minor/major faults has not been seen in the literature of reliability modeling. In addition to the above idea, the boiler of the Steam Generation Plant studied, in which the type of service provided for a boiler depends upon major and minor faults. Minor faults are repaired easily while major faults are replaceable.
Initially, a three-unit system with a boiler and two FD fans has considered for the study. When a boiler fails system stops working immediately, but if any one of the FD fans fails system goes on reduced capacity. For continuous functioning, a boiler and 2 FD fans should function. On boiler failure, two possibilities arise major and minor faults. Repair facility provided for a small crack in the outer chamber of the boiler that occurs due to overheating & erosion, whereas the repairman performed replacement for the major equipment failures for a boiler. Priority of repair is given to the boiler over fans so that the system can operate for a long time. For the FD fans, repair priority is on an FCFS basis.
The following reliability measures were computed numerically using semi-Markov processes and regenerative point techniques and also plotted graphically based on the information gathered from the industry :
• Mean time to system failure (MTSF).
• Availability analysis at full capacity.
• Availability analysis at reduced capacity.
• Busy period for repair time only.
• Busy period for replacement time only.
• Expected no. of repairs.
• Expected no. of replacements.
• Cost-benefit analysis.
II. Model Description I. State Transition Diagram
Figure 1, shows the state transitions diagram of the steam generation plant consisting of one boiler and two FD fans.
Figure 1: State Transition Diagram
Table 1: State Discription
States Discription
So This is the operating state of the system
So, Si, S2, S3, S4, S5, S6, S8, S9 The epoch of entry into these states are regenerative points thus,
these states are called regenerative states
S3, S4 These are reduced capacity states
Si, S2, S5, S6, S7, S8, S9, S10 These are failed states of the system.
HI Assumptions
The stated model follows these assumptions.
• All the random variables are independent.
• Failure times distribution is exponential, whereas repair times have distributed arbitrarily.
• System works as well as new after every repair.
• when the failure occurs repairman will come immediately.
III. Nomenclature
Table 2: Notations & symbols for the states of system
Notations
Notations Discription
A Constant failure rate of Boiler.
Ai Constant failure rate of FD fan i.
A2 Constant failure rate of FD fan 2.
a Repair rate of Boiler.
ai Repair rate of FD fan i.
a2 Repair rate of FD fan 2.
Y Replacement rate of Boiler.
p probability of minor failure in Boiler.
q probability of major failure in Boiler.
G(t), g(t) c.d.f. & p.d.f of repair time of Boiler.
H(t), h(t) c.d.f. & p.d.f of replacement time of Boiler.
Gi(t), gi(t) c.d.f. & p.d.f of repair time of FD fan i.
G2(t), g2(t) c.d.f. & p.d.f of repair time of FD fan 2.
Symbols for the states of the system
Symbols Discription
St states of the system, i = i, 2,3,..., i0
Bo, FDio, FD20 Boiler, FD fans i and 2 are in operating state respectively.
Br Boiler under repair.
Brep Boiler under replacement.
Bs, FDis, FD2s Boiler, FD fans i and 2 in standby state.
FDir, FD2r FD fans i and 2 under repair.
FDiwr, FD2wr FD fans i and 2 are waiting for repair.
FDir, FD2R FD fans i and 2 are under repair from previous state.
IV. Transition Probabilities & Mean Sojourn Times
The pij represents non-zero elements which are given below
poi =
po3 = pio =
p35 =
/IfJ
A + Ai + Xi'
A + Ai + A2 P20 = 1,
P02
P04
A
A + Ai + /2' A2
A + Ai + Ai P30 = gi* (A + Ai),
A p
A + A2
[i - gi*(A + A2)],
P36 =
Aq
A + A2
[i - gi * (A + A2 )],
P37
Л2
л + я2
[1 - g!* (л + л2)],
[1 - gl * (Л + Л2)],
Р40 = g2* (Л + Л1),
л^
Р79 = [1 - g2* (Л + Л1)],
,(10)
Л + Л1 Л1
Р78
Р7,10
Л + Л2
Л p Л + Л1
Л1 Л + Л1
[1 - g2* (Л + Л1)], [1 - g2* (Л + Л1)],
Р71з0) = ^ [1 - g2* (Л + Л1)],
Р53 = 1,
Р63 = 1, Р74 = 1,
Р84 = 1, P94 = 1,
P10,3 = 1
It can be verified by these probabilities that
P01 + P02 + P03 + P04 = 1, P10 = 1,
P30 + P35 + P36 + P37 = 1, P30 + P35 + P36 + Рм = 1, Р40 + Р48 + Р49 + p430) = 1, Р53 = Р63 = Р74 = 1,
Also Ц1, the mean sojourn times in state Sj are
P20 = 1,
p70 + p78 + p79 + p7,10 = 1, p87 = p97 = p10,3 = 1
Ц0
1
Ш = -g*' (0), Ц2 = -h*' (0), Ц3
1
Л + Л2
Л + Л1 + Л2' 1
И7 = ^^[1 - g2* (Л + Л1)], Ц5 = -g*'(0), И6 = -h*'(0), Ц7 = -g1*'(0),
[1 - g1* (Л + Л2)],
Л + Л11 № = -g*' (0),
Ц9 = -h*' (0), Ц10 = -g2*' (0),
The unconditional mean time taken by the system to transit for any regenerative state j' when it (time) is counted from the epoch of entrance into state 'i' is mathematically represented as
m
ij
!• TO
Уо tdQij(t) = -qij*'(0)
where
m01 + m02 + m03 + m07 = Ц0, m20 = ^2,
(7)
m30 + m35 + m36 + m37 = ^3 + K1, m70 + m78 + m79 + m7^0) = ^7 + K2, m63 = ^6, m87 = ^8, m10,3 = Ц10
K1 = Г fg1 (t)dt, K2 = £/0 ^ ig2(t)dt,
m10 = ^1,
m30 + m35 + m36 + m37 = ^3, m70 + m78 + m79 + m7,10 = щ, m53 = ^5, m77 = Ц7, m97 = ^9,
Л1
Л
(1)
III. Reliability Measures for System Effectiveness I. Mean Time to System Failure (MTSF)
When the system starts from the initial state So, Mean time to system failure (MTSF) of the system is determined by considering failed state as absorbing state as given below
MTSF = T0 = lim
s^0
1 - Ф0** (s)
s
Using L' Hospital Rule & putting the value of <0** (s), we have
N
To = d (3)
where
N = ^o + № po3 + m po4 D = 1 - po3 p30 - p04 p40
II. Availability Analysis at Full Capacity
Using the theory of regenerative processes, the availability AF0 of the system at full capacity is given by
Afo = lim(sAfo* (s)) = Nl
s^0 Di
where
Ni = ^0 [(1 - p48 - p49)(1 - p35 - p36) - p34)p430)] (4)
& Di = ^o[p40 - p30p43 - p40p35p53 - p40p63p36] + (^1 poi + ^2p02)[(1 - p48p84 - p49p94) (1 - p36p63 - p35p53) - p43p34] + [^3 + K1] [po3 + p04p43 - p03p48p84 - p03p49p94] + [^4 + K2] [po4 + p03p34 - p04p35p53 - p04p36p63] + (^5p35 + ^6p36)[(1 - p01 p10 - p02p20)(1 - p48p84 - p49p94) - p04p40]
+ (№p48 + ^9p49)[(1 - p01 p10 - p02p2o)(1 - p35p53 - p63p36) - p03p30] (5)
III. Availability Analysis at Reduced Capacity
Using the theory of regenrative processes, the availability AR0 of the system at reduced capacity is given by
N2
Aro = lim(sAro * (s)) = -2
s^0 D1
where
N2 = U3 [p03 (1 - p48 - p49) + p04p43,0)] + p04(1 - p35 - p36) + p03p^ ] (6)
and D1 is already specified in equation (5).
IV. Busy Period for Repair Time only
In steady state, busy period for repair time is defined as the time for which system is under repair by repairman and is given by
N3
Bro = lim(sBr0* (s)) = -3 (7)
s^0 D1
Where
N3 = U1 po1[(1 - p48 - p49)(1 - p35p36) - p43(10)p34(7)] + ^03(1 - p48 - p49) + p04p43(10)]
+ ^4[po4(1 - p35 - p36) + p03p34(7)] + V5p35[p03p40 + (p03 + p04)p43(10)]
+ ^8p48[p04(1 - p35 - p36) + p03p34(7)] (8)
& D1 is already specified in equation (5).
V. Busy Period for Replacement Time only
In steady state, busy period for replacement time is defined as the time for which system is busy under replacements and is given by
Brpo = lim(sBRpo * (s)) = N (9)
s^0 U\
where
N4 = mP02[(1 - ¡48 - ¡49)(1 - ¡35¡36) - ¡43(10)p34(7)] + H¡36[¡03(1 - ¡48 - ¡49) + p43(10)]
+ F9¡49 [¡04(1 - ¡35 - ¡36) + ¡34(7)] (10)
& U1 is already specified in equation (5).
VI. Expected No. of Repairs
Expected number of repairs per unit time for the system is given by
Vr0 = lim[sVr0** (s)] = N (11)
where
N5 = (1 - ¡02)(1 - ¡35 - ¡36)(1 - ¡48 - ¡49) - (1 - ¡02)¡34(7)¡430 + ¡03¡48¡34 + ¡04¡35¡430
+ ¡04¡48(1 - ¡35 - ¡36) + ¡03¡35(1 - ¡48 - ¡49) (12)
& U1 is already specified in equation (5).
VII. Expected No. of Replacements
Expected number of replacements per unit time for the system is given by
Vrp0 = lim[sVrp0 ** (s)] = N (13)
s^0 U\
where
N6 = ¡02((1 - ¡35 - ¡36)(1 - ¡48 - ¡49) - ¡34(7)¡4^0)) + ¡03(¡36(1 - ¡48 - ¡49) + ¡49¡34 )
+ ¡04(¡49(1 - ¡35 - ¡36) + ¡36¡430)) (14)
& U1 is already specified in equation (5).
IV. Cost-Benefit Analysis
The expected total profit incurred to the system is
P0 = C0AF0 + C1ar0 - C2BR0 - C3BRP0 - C4 vr0 - C5VRP0
where
C0 = revenue per unit up time at full capacity
C1 = revenue per unit time at reduced capacity
C2 = cost per unit time when repairman is busy in doing repair
C3 = cost per unit time when repairman is busy in doing replacement
C4 = cost per repair.
C5 = cost per replacement.
V. Particular Cases
For the numerical evaluation and graphical plotting for various reliability measures, the following particular cases are considered.
Let us assume that g(t) = ae-at, h(t) = ye-Yt,g1 (t) = a1 e-a1t,g2(t) = a2e-a2t, and the remaining distributions are the same as in the general case. Therefore, we have
Poi
P04 P30 P37 P49
A p
A + Ai + A2 ' A2
A + Ai + A2 ai
A + ai + A2' A2
A + ai + A2 Aq
= P34
(7)
A + a2 + Ai'
P84 = P94 = Pi0,3 = i, i
U2
U8
7 i
a' i
a'
P02
Aq
A + Ai + A2 ' Pio = i,
AP
P35 P40 P4,i0
^9
A + ai + A2' a2
A + Ai + a2 Ai = A + a2 + Ai i
A + Ai + A2 ' i
A + ai + A2' i
y' i
Y'
P43
(i0)
P03
Ai
A + Ai + A2' P20 = i
Aq
P36 P48
A + ai + A2' AP
A + a2 + Ai'
P53 = P63 = P74 = i i
V4 H7 Hi0
i
A + a2 + Ai' i
ai i
a2
Table 3: ComPutation of various rates/costs on the basis of actual data collected from industry
Various rates/ cost associated
corresponding values
Failure rate of Boiler (A) Failure rate of FD fan i (Ai) Failure rate of FD fan 2 (A2) Repair rate of Boiler (a) Replacement rate of Boiler (7) Repair rate of FD fan i (ai) Repair rate of FD fan 2 (a2) Expected cost per repair (C4) Expected cost per replacement (C5)
0.000ii86/hr 0.000ii7i/hr 0.000i0i295/hr 0.00738/hr 0.0008733 /hr 0.024272/hr 0.048544/hr Rs. i4282 Rs. i579627
Hypothetical values have been taken for remaining rates/costs. arious reliability measures for system performance have been computed in table 4 by putting the values given in the table 3 based on particular cases.
Table 4: Computation of various measures of system effectiveness
Mean time to system failure 08376.82/hr
Availability of the system at full capacity 0.344986/hr
Availability of the system at reduced capacity 0.639788/hr
Busy period of repairman for repair time only 0.644142/hr Busy period of repairman for replacement time only 0.010379/hr
Expected no. of repairs 0.000107/hr
Expected no. of replacements 0.000009/hr
a
VI. Results and Discussion
Figure 2: MTSF vs Failure rate of Boiler
Figure 3: Profit vs Revenue up-time for the system Table 5: Cut Point for profit w.r.t. Revenue Up-time of the system.
Failure rate of FD fan Revenue per unit up time (Rs.) Profit (Rs.) one (/hr)
A\ = .0001171 Co < or = or > 731316.24 negative or zero or positive
A\ = .0001571 C0 < or = or > 826502.352 negative or zero or positive
At = .0002171 C0 < or = or > 969279.171. negative or zero or positive
In figure 2, the effect of the failure rate of Boiler(A) on MTSF has shown for the different values of the failure rate of FD fan one (Ai). As the failure rate (A) increases, the MTSF of the system decreases. Also, as the failure rate (A1) increases MTSF of the system decreases. In figure 3, the effect of cost per unit up time of the system (C0) w.r.t. profit has shown for the different
values of failure rate of FD fan one (A1). As the cost C0 increases, profit of the system increases. Also, as the failure rate of FD fan one (Ai) increases profit decreases. Various cut point formed from the graph of profit w.r.t. revenue Up-time of the system as shown in table 5.
VII. Conclusion
Reliability modeling established for steam generation plant that shows the effect of service type & type of failures on system performance. The study reveals that the busy period of repairman for repair time is more as compared to replacement time. Also, cut-off points formed from profit helps the industrialist to maintain the economy of their system. In addition, any industry can consider the stated model to enhance the performance of their system using the different rates for repair/replacement.
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