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Reliability analysis of the naphtha fuel oil system in a thermal power plant
Microelectron.Reliab.,Vol. 34, No. 2, pp. 369-372, 1994. Printed in Great Britain.
0026-2714/9456.00 + .00 © 1993 Pergamon Press Lid
TECHNICAL NOTE RELIABILITY ANALYSIS OF THE NAPHTHA FUEL OIL SYSTEM IN A THERMAL POWER PLANT SUDHAKARKAUSHIK and I. P. SINGH Mechanical Engineering Department, Regional Engineering College, Kuruksbetra 132 119, India
(Received for publication 6 August 1992) Abstract--The paper discusses the performance and reliability analysis of the naptha fuel oil system in a thermal power plant. Reliability, the availability function (Av) and the mean time to system failure (MTSF) of this system have been calculated. An example follows, assuming various values of failure and repair rate.
INTRODUCTION This paper deals with the reliability analysis of the naphtha fuel oil system in a thermal power plant. The naphtha fuel oil system carries the oil from oil tanks to the gas turbine combustors. High reliability of this system will ensure that the gas turbines remain operative and hence there is continuous power production. In this paper, priority in repair is given to the component which works at 100% capacity, rather than the standby component which works at 30% capacity when both of them are in a failed state. The reliability analysis here provides us with the value of the availability function, reliability and mean time to system failure in terms of the failure rate and repair rate of different components. Failure and repair rates are taken to be constant. Similar systems in other industrial fields can also be analysed with the help of this reliability analysis. SYSTEM DESCRIPTION The thermal plant has a naphtha fuel oil system which carries the naphtha fuel from the naphtha fuel oil tank to the turbine combustors. The system has the following subsystems: (i) fuel pumping system A with three units in series (combination of suction oil strainer, fuel oil transfer pump and main fuel forwarding pump, for the filtration of fuel oil first and then to pump it to the relevant valves) so that failure of any one causes the failure of subsystem A ; (ii) the standby pumping system B similar to system A, with three units in series. System B works when A is in a failed state. The failure of both
(iv) fuel oil valve D - - a standby for C. The capacity of fuel oil valve D is less than the capacity o f C (the capacity of D is about 30% of that of C). So the failure of C makes valve D operative but the whole system works at reduced capacity. Failure of both C and D leads to failure of the whole system. ASSUMPTIONS AND NOTATIONS (i) (ii) (iii) (iv)
Failure and repair rates are constant; failure and repair are statistically independent; a repaired unit is as good as new; if C and D both fail then the repair of C will be carried out first as D is of a lesser capacity. Also if A and C (or B and C) fail, then C will be repaired first as replacement of C by the standby leads the system to having a reduced capacity. But if A, B and C fail, then A and B will be repaired first so as to make the system operative; (v) the repair of A and B can proceed simultaneously and i r a and D (or B and D) fail, then the repair of A (or B) will be performed first. The following symbols are associated with the system: A, B, C, D units in an operating state ~1, :B, U, D units in a failed state 2~, (7~ constant failure rate of ith components in A and B 2 = ~ 2, cr = y. ~, /~,, #~ constant repair rate of ith component in A and B (i = 1 , 2 , 3 )
/~ /~ == ~~#/ ~~ ~, ~, constant failure rate of C and D, respectively ~, ~ constant repair rate of C and D, respectively P ~ ( t ) the probability that at time t all units are good and the system is working at a full capacity, where A, B, C, D are replaced by ~, ~, ~', D for their respective failure s Laplace transform variable. The transition diagram for the system is shown in Fig, I,
A and B leads to the failure of the whole system; (iii) fuel oil valve C through which fuel oil goes to turbine combustors; 369
370
Technical Note
System working at full capacity System working at reduced capacity
IX
I
I
System in failed state
Fig. 1. __d P~o(t) + (E + 2 + ~ ) P ~ ( t )
ANALYSIS OF THE SYSTEM
The differential equations associated with the transition diagram are:
dt
d P~)(t) + (2 + c~)P~)(t) dt
= tJe~(t) + EPg(t) + tIPc~+itPcn, AB ~
(1)
= c~P~(t) + flP~(t),
(7)
d P ~ ( t ) + EPiC(t) = 7 P ~ ( t ) + ~PA~(t), at
(8)
d PoD(t) ~ + (# + p)P~g(t) = .P~g(t) + ).P~g(t),
d pacg(t ) + (a + fl + ot)P~c~(t) dt
(9)
d P ~ ( t ) + EPiC(t) = eP~(t) + ~P~c~(t), (10) dt
= 2 P g ( t ) + #P~cg(t) + EP~cg(t), (2) d PcD(t) ~B + (it + fl)P~c~D(t)= aP~c~)(t)+ 2P~g(t), dt
d P~c~(t) + (ca + #)P~c~(t) = EPiC(t) + 2 P ~ ( t ) , "dt
(3)
d Pg~(t) + (It + 2 + a)eA~o(t) dt = flP~c~o(t)+EP~D(/),
(ll)
d Pgg(t) + (c~ + It)P~g(t) = EP~g(t) + aPA~(t), dt (4)
(12) d
d e~)(t)+(E + a +T)e~g(t) dt
=~fcg(t)+uP~g(t)+2P~(t), deg(t)+(2
+E +7)Pg(t)=~P~g(t),
dtP~+~e~=yP~g(t)+ctP~(t), (5)
Table 1 ~t
a
),
2=0.02
d
~Pg~(t)+(~t + 2 + ~ +~)P~(t) =~Pg~(t)+#P~c~)(t)+uP~g(t).
(6)
dt
Availability (A~) ).=0.04 2=0.06
0.05 0.03 0.002 0.9474279 0.9452848 0.9428386 0.07 0.03 0.002 0.9352532 0.9321569 0.9296985 0.05 0.04 0.002 0.9463178 0.9439234 0.9413638
(13)
(14)
With the initial condition P~o~(0)= 1, otherwise it equals zero. Using the Laplace transform technique the probability tranforms are given by: P~)(s)=
I
s+2+~-/~
P~c~(S)=~P~(s),
T2
~:t
M:
P~o(s)=T~P~)(s),
1-1
Technical Note
T6Pco(S)'
371
Ml=
s+a
+ 2 + ( r +r/
eg(s)= r,?~(s), eg(s)=T~e¢o(s)," P ~ ( s ) = ~ paat.~ cow,,
M2
~
P~g(s) = M , P ~ ( s ) .
e ~ ( s ) =-~1 e ~ ( s ) ,
e~(s) = M6P~(s),
P~g(s) = M3P~g(s),
P~g(s) = MTP~g(s),
M2 = - ~ + - ~ - - ~ 8
- ErT~
M4=
T7
~
M5 = -~'~6 +
K=(, +~ +,+~,),
a n~l ~-E, + ~ , J ,
-YT5
K2= ($ ~_. _~, _ ~i), K3 = ( K - [ 3 2
MT= ~ + ~6M6] • Setting d/dt=O, P ~ ( t ) steady state probabilities:
,
K4 =
s + /~ + 2 + ~ -- E,,
]
~M,,
-EFT5
M,=
K t = ( s +E + a + V), a#
~m, 1
~-~s+K-----~J,
e~f~(s) = M4PcD(S), "" where
~ + - ~ 7 T' ..i ,
et M21
- ~0~
M3=
~
P~cf = ~
= (s + e), s+~+~
)
Ks =
s+a+,8
~:~ ,
#a K,
L.
e#~t
e#[3tr~t2
K~K2 K~K~K3 K2K3K4K,
[30"220~
AB
~
AB
an P~c~ -A~ - G2Pco, P~g=Gt
pA[3oeo~ ~.o~ K,K~K3K4K ~ K-~ where
K~K2K3K ~ ~ 4
V=(2+~+?),
, Kd K, ~K-~- -, ~ -K,
V2 = ~, + [3 -- E V4=
£[3¢7,~,2a 2
E, = I_a + [3 + a
epffa T 2 "+ _ _ .l , T3-+ K,K~K3K, T, KK, K2K3K, j
fla2a ]
'
Vj=(*+a+~),
V3=
'
~-~-~-~0~--~3 [
KK, K[----~ + KK~ K~ K 3K4
KK,K~K3I~K,J-
Aa P ca~~ -_ G~Pco, A~ _ A~ eev-Gseco,
_ a~ e~g-G3eco)
e 2#Affair2
T3= L~, ~ + K I K ~ 4 K J ~ ÷ r4=
havo tho
P ~ - G__P c DAB ,
T2= 24 KK, K2K3K, K +-K---~+ KK, K ~ +
o,.,
AB _ Otpan P r o - V co,
K7=
T~ = Fs + a + [3 + 2
A.
~o A D ~ ~ ~z':'5~CD~ oAB
co,
_
K,
K, ~
,
Vs-~- ~-J~[3 fl,~.£[30"a
I"5
V l V2 V3 V, I"5
V2V3Z4V5
[3a22a ¢.,~20~2 ÷ VV, V~--------~3~ VV~ V~ V3 V,
T6-- L K - ~ y + K---fi~,K~+ E~ 7",
+ VVll~2gt2I~2v;-2 v3 ),4
'
~[3
Vl "~'~2V2V3 V4 V5 J ,
VV, V2V3V, Vs+-V--~ VV, V2
[a T: [3aa r~ = ~ r, -~ 7", K, K,----~3~ KK, K~------~3 ' ,
[3). '
I.~O
V2V2ViV4
+
V
'
Vl V 2 V 3 V 4 V s J e j
"~ V V I ~ 2 V 3 V 4 V s j
)
372
Technical Note
Ifl
~flo'~t E2
E4 =
~flo',~.~t 1
where
E3 ~- VI V2 V3 V4 EI ~" VVI V2 V3 V4 '
H= I+-~j+E3+E4+Es+E6+E 7
g5= V3 4-I F ~
~-- L~
VIV2V3EI
E2
VVlV2V3J'
~7,~Ot
r, + ~
,~
1
"
+-~ + G + G1+ G2 -F G3 + G4 + Gsj •
1
÷ ~ ~,J,
The Laplace transformof the reliability functionR(s) is given by
~7 = L ~ E,, +V, e, + ~
,
R(s) = ?~g(s) + e~g(s) + P~g(s) + e~g(s) + e~(s) + ,"~g(s) + e~g(s)
-~ + E7 + E, ,
+ Pg~(s) + P~g(s),
~[' ]
1"2 M2 R(s)= I+'-~I+T4+T5+-~I+K+T7
c,=# -6+c ,
The availability function Av is given by: G~ =
E, +-~ a~
,
Ao = e ~
L
Using the normalizing condition: AB A~
Pe~ + Pc~ A~ AB + Pc~ + Pe~ =
Pc~ + Pe~ 1.
A~
A~
A,
_ei
The mean time to system failure is given by: M T S F = lim R(s)= -H1I! ~,Ez ~a+pas
s~o
+_ + E4 + Es + G
,
] EXAMPLE
We have
PcoAn=
+ egg +
[H]-~,
Assuming fl = 0.25, # = 0.2, ~ = 0.3 and ~/= 0.03, the values of availability function A v are tabulated in Table 1 for various values of 2, ~t, tr, ?. The table shows the decrease in availability with the increase in failure rates.
0026-2714/9456.00 + .00 © 1993 Pergamon Press Lid
TECHNICAL NOTE RELIABILITY ANALYSIS OF THE NAPHTHA FUEL OIL SYSTEM IN A THERMAL POWER PLANT SUDHAKARKAUSHIK and I. P. SINGH Mechanical Engineering Department, Regional Engineering College, Kuruksbetra 132 119, India
(Received for publication 6 August 1992) Abstract--The paper discusses the performance and reliability analysis of the naptha fuel oil system in a thermal power plant. Reliability, the availability function (Av) and the mean time to system failure (MTSF) of this system have been calculated. An example follows, assuming various values of failure and repair rate.
INTRODUCTION This paper deals with the reliability analysis of the naphtha fuel oil system in a thermal power plant. The naphtha fuel oil system carries the oil from oil tanks to the gas turbine combustors. High reliability of this system will ensure that the gas turbines remain operative and hence there is continuous power production. In this paper, priority in repair is given to the component which works at 100% capacity, rather than the standby component which works at 30% capacity when both of them are in a failed state. The reliability analysis here provides us with the value of the availability function, reliability and mean time to system failure in terms of the failure rate and repair rate of different components. Failure and repair rates are taken to be constant. Similar systems in other industrial fields can also be analysed with the help of this reliability analysis. SYSTEM DESCRIPTION The thermal plant has a naphtha fuel oil system which carries the naphtha fuel from the naphtha fuel oil tank to the turbine combustors. The system has the following subsystems: (i) fuel pumping system A with three units in series (combination of suction oil strainer, fuel oil transfer pump and main fuel forwarding pump, for the filtration of fuel oil first and then to pump it to the relevant valves) so that failure of any one causes the failure of subsystem A ; (ii) the standby pumping system B similar to system A, with three units in series. System B works when A is in a failed state. The failure of both
(iv) fuel oil valve D - - a standby for C. The capacity of fuel oil valve D is less than the capacity o f C (the capacity of D is about 30% of that of C). So the failure of C makes valve D operative but the whole system works at reduced capacity. Failure of both C and D leads to failure of the whole system. ASSUMPTIONS AND NOTATIONS (i) (ii) (iii) (iv)
Failure and repair rates are constant; failure and repair are statistically independent; a repaired unit is as good as new; if C and D both fail then the repair of C will be carried out first as D is of a lesser capacity. Also if A and C (or B and C) fail, then C will be repaired first as replacement of C by the standby leads the system to having a reduced capacity. But if A, B and C fail, then A and B will be repaired first so as to make the system operative; (v) the repair of A and B can proceed simultaneously and i r a and D (or B and D) fail, then the repair of A (or B) will be performed first. The following symbols are associated with the system: A, B, C, D units in an operating state ~1, :B, U, D units in a failed state 2~, (7~ constant failure rate of ith components in A and B 2 = ~ 2, cr = y. ~, /~,, #~ constant repair rate of ith component in A and B (i = 1 , 2 , 3 )
/~ /~ == ~~#/ ~~ ~, ~, constant failure rate of C and D, respectively ~, ~ constant repair rate of C and D, respectively P ~ ( t ) the probability that at time t all units are good and the system is working at a full capacity, where A, B, C, D are replaced by ~, ~, ~', D for their respective failure s Laplace transform variable. The transition diagram for the system is shown in Fig, I,
A and B leads to the failure of the whole system; (iii) fuel oil valve C through which fuel oil goes to turbine combustors; 369
370
Technical Note
System working at full capacity System working at reduced capacity
IX
I
I
System in failed state
Fig. 1. __d P~o(t) + (E + 2 + ~ ) P ~ ( t )
ANALYSIS OF THE SYSTEM
The differential equations associated with the transition diagram are:
dt
d P~)(t) + (2 + c~)P~)(t) dt
= tJe~(t) + EPg(t) + tIPc~+itPcn, AB ~
(1)
= c~P~(t) + flP~(t),
(7)
d P ~ ( t ) + EPiC(t) = 7 P ~ ( t ) + ~PA~(t), at
(8)
d PoD(t) ~ + (# + p)P~g(t) = .P~g(t) + ).P~g(t),
d pacg(t ) + (a + fl + ot)P~c~(t) dt
(9)
d P ~ ( t ) + EPiC(t) = eP~(t) + ~P~c~(t), (10) dt
= 2 P g ( t ) + #P~cg(t) + EP~cg(t), (2) d PcD(t) ~B + (it + fl)P~c~D(t)= aP~c~)(t)+ 2P~g(t), dt
d P~c~(t) + (ca + #)P~c~(t) = EPiC(t) + 2 P ~ ( t ) , "dt
(3)
d Pg~(t) + (It + 2 + a)eA~o(t) dt = flP~c~o(t)+EP~D(/),
(ll)
d Pgg(t) + (c~ + It)P~g(t) = EP~g(t) + aPA~(t), dt (4)
(12) d
d e~)(t)+(E + a +T)e~g(t) dt
=~fcg(t)+uP~g(t)+2P~(t), deg(t)+(2
+E +7)Pg(t)=~P~g(t),
dtP~+~e~=yP~g(t)+ctP~(t), (5)
Table 1 ~t
a
),
2=0.02
d
~Pg~(t)+(~t + 2 + ~ +~)P~(t) =~Pg~(t)+#P~c~)(t)+uP~g(t).
(6)
dt
Availability (A~) ).=0.04 2=0.06
0.05 0.03 0.002 0.9474279 0.9452848 0.9428386 0.07 0.03 0.002 0.9352532 0.9321569 0.9296985 0.05 0.04 0.002 0.9463178 0.9439234 0.9413638
(13)
(14)
With the initial condition P~o~(0)= 1, otherwise it equals zero. Using the Laplace transform technique the probability tranforms are given by: P~)(s)=
I
s+2+~-/~
P~c~(S)=~P~(s),
T2
~:t
M:
P~o(s)=T~P~)(s),
1-1
Technical Note
T6Pco(S)'
371
Ml=
s+a
+ 2 + ( r +r/
eg(s)= r,?~(s), eg(s)=T~e¢o(s)," P ~ ( s ) = ~ paat.~ cow,,
M2
~
P~g(s) = M , P ~ ( s ) .
e ~ ( s ) =-~1 e ~ ( s ) ,
e~(s) = M6P~(s),
P~g(s) = M3P~g(s),
P~g(s) = MTP~g(s),
M2 = - ~ + - ~ - - ~ 8
- ErT~
M4=
T7
~
M5 = -~'~6 +
K=(, +~ +,+~,),
a n~l ~-E, + ~ , J ,
-YT5
K2= ($ ~_. _~, _ ~i), K3 = ( K - [ 3 2
MT= ~ + ~6M6] • Setting d/dt=O, P ~ ( t ) steady state probabilities:
,
K4 =
s + /~ + 2 + ~ -- E,,
]
~M,,
-EFT5
M,=
K t = ( s +E + a + V), a#
~m, 1
~-~s+K-----~J,
e~f~(s) = M4PcD(S), "" where
~ + - ~ 7 T' ..i ,
et M21
- ~0~
M3=
~
P~cf = ~
= (s + e), s+~+~
)
Ks =
s+a+,8
~:~ ,
#a K,
L.
e#~t
e#[3tr~t2
K~K2 K~K~K3 K2K3K4K,
[30"220~
AB
~
AB
an P~c~ -A~ - G2Pco, P~g=Gt
pA[3oeo~ ~.o~ K,K~K3K4K ~ K-~ where
K~K2K3K ~ ~ 4
V=(2+~+?),
, Kd K, ~K-~- -, ~ -K,
V2 = ~, + [3 -- E V4=
£[3¢7,~,2a 2
E, = I_a + [3 + a
epffa T 2 "+ _ _ .l , T3-+ K,K~K3K, T, KK, K2K3K, j
fla2a ]
'
Vj=(*+a+~),
V3=
'
~-~-~-~0~--~3 [
KK, K[----~ + KK~ K~ K 3K4
KK,K~K3I~K,J-
Aa P ca~~ -_ G~Pco, A~ _ A~ eev-Gseco,
_ a~ e~g-G3eco)
e 2#Affair2
T3= L~, ~ + K I K ~ 4 K J ~ ÷ r4=
havo tho
P ~ - G__P c DAB ,
T2= 24 KK, K2K3K, K +-K---~+ KK, K ~ +
o,.,
AB _ Otpan P r o - V co,
K7=
T~ = Fs + a + [3 + 2
A.
~o A D ~ ~ ~z':'5~CD~ oAB
co,
_
K,
K, ~
,
Vs-~- ~-J~[3 fl,~.£[30"a
I"5
V l V2 V3 V, I"5
V2V3Z4V5
[3a22a ¢.,~20~2 ÷ VV, V~--------~3~ VV~ V~ V3 V,
T6-- L K - ~ y + K---fi~,K~+ E~ 7",
+ VVll~2gt2I~2v;-2 v3 ),4
'
~[3
Vl "~'~2V2V3 V4 V5 J ,
VV, V2V3V, Vs+-V--~ VV, V2
[a T: [3aa r~ = ~ r, -~ 7", K, K,----~3~ KK, K~------~3 ' ,
[3). '
I.~O
V2V2ViV4
+
V
'
Vl V 2 V 3 V 4 V s J e j
"~ V V I ~ 2 V 3 V 4 V s j
)
372
Technical Note
Ifl
~flo'~t E2
E4 =
~flo',~.~t 1
where
E3 ~- VI V2 V3 V4 EI ~" VVI V2 V3 V4 '
H= I+-~j+E3+E4+Es+E6+E 7
g5= V3 4-I F ~
~-- L~
VIV2V3EI
E2
VVlV2V3J'
~7,~Ot
r, + ~
,~
1
"
+-~ + G + G1+ G2 -F G3 + G4 + Gsj •
1
÷ ~ ~,J,
The Laplace transformof the reliability functionR(s) is given by
~7 = L ~ E,, +V, e, + ~
,
R(s) = ?~g(s) + e~g(s) + P~g(s) + e~g(s) + e~(s) + ,"~g(s) + e~g(s)
-~ + E7 + E, ,
+ Pg~(s) + P~g(s),
~[' ]
1"2 M2 R(s)= I+'-~I+T4+T5+-~I+K+T7
c,=# -6+c ,
The availability function Av is given by: G~ =
E, +-~ a~
,
Ao = e ~
L
Using the normalizing condition: AB A~
Pe~ + Pc~ A~ AB + Pc~ + Pe~ =
Pc~ + Pe~ 1.
A~
A~
A,
_ei
The mean time to system failure is given by: M T S F = lim R(s)= -H1I! ~,Ez ~a+pas
s~o
+_ + E4 + Es + G
,
] EXAMPLE
We have
PcoAn=
+ egg +
[H]-~,
Assuming fl = 0.25, # = 0.2, ~ = 0.3 and ~/= 0.03, the values of availability function A v are tabulated in Table 1 for various values of 2, ~t, tr, ?. The table shows the decrease in availability with the increase in failure rates.