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Design and cost analysis of a refining system in the sugar industry
Microelectron. Reliab., Vol. 30, No. 6, pp. 1025-1028, 1990. Printed in Great Britain.
0026-2714/9053.00+ .00 © 1990PergamonPress plc
DESIGN A N D COST ANALYSIS OF A REFINING SYSTEM IN THE SUGAR INDUSTRY DINESH KUMAR,* JAI SINGHt *Mechanical Engineering Department and ?Mathematics Department, Regional Engineering College, Kurukshetra--132119 (Haryana), India and P. C. PANDEY Mechanical and Industrial Engineering Department, University of Roorkee, Roorkce (U.P.), India
(Received for publication 17 November 1989) Abstract--The role of reliability technology in the sugar industry is discussed. The industry uses various systems, viz. feeding, crushing-cum-filtering,heating-cum-sulphonationand cleaning, cooking, crystallization, separation-cure-cooling and collection for storage. We consider only the refining system, the main functionary part of the sugar industry, which consists of four subsystems A, B, D and E. The failure rate for each subsystem is taken to be constant, whereas the repair rates are variable. The analysis is carried out using a supplementary variable technique. A particular case and a profit-analysis are discussed.
INTRODUCTION In a process industry, to obtain maximum output it is necessary to run each of the units in good condition, i.e. each piece of equipment of the unit should run failure free. For this purpose, a repair facility is provided, depending upon the cost involved and the resources available. Reliability analysis helps in coordinating the failure behaviour of each piece of equipment and the repair possibility within a stipulated time with m i n i m u m loss in production. In the sugar industry (a process industry), we may use reliability technology successfully to give maximum production and to maintain quality. In the sugar industry, the juice from a feeding system [1] containing small pieces of bagasse (i.e. refuse after extracting juice from cane sugar) and m u d is subjected to filtration. The filter consists of units in series to ensure the complete removal of bagasse from juice. The bagasse-free juice is diluted with water to increase its fluidity and is sent to the heating unit. This unit is similar to a heat exchanger to exploit the maximum a m o u n t of heat produced in the system. Steam at a high temperature warms the juice and, being cooled in the process, is sent back to the boiler. The juice remains in the heater for a certain period to achieve a given pH value. Heated juice with the required pH value is sent to the sulphonation unit, where sulphur dioxide is passed through it to remove the mud. The juice is again passed through the heater. This heated juice is then sent to the clarifier where the remaining cane mud is removed by settling. The clarifier has a vibration mechanism. The heated mixture is stored in the vibrating cavity of the clarifier, and the clear juice
which separates at the upper surface is then collected in the tank. The unit has a filter with small holes which are frequently blocked by cane mud, so regular washing is provided. THE SYSTEM The system has four subsystems, A, B, D and E, as follows: (1) The filter (A) consists of two units in series. Failure of any one causes complete failure of the system. (2) The ratifier (B) consists of n units in series. Failure of any one causes complete failure of the system. (3) The sulphonation plant (D) consists of l units in parallel. Failure of units reduces the capacity of the system and hence loss in production. Complete failure occurs only when all units fail. (4) The heating plant (E) consists of m units in parallel. Failure of units reduces the capacity of the system. Complete failure occurs only when all units fail. ASSUMPTIONS (1) Failure rates for the system are constant and statistically independent. (2) Intermittent service is possible. (3) Service includes repair and/or replacement. (4) Equipment can be replaced on failure or at the time of first service call after a specified period, whichever comes first. (5) Repair rates in each subsystem are arbitrary. (6) Each subsystem has a separate repair thcility. There is no waiting time for repair in the system. (7) A repaired unit is as good as new. NOTATION
A,B,D,E
I025
indicate that units are in full operating state; a bar over upper-case letters represents reduced state, whereas lower-case letters indicate failed state
DINESH KUMAR et al.
1026
"l i = 1, 2, 3, 4 rate for complete failure of subsystems A, B, D and E respectively 2~, 22 transition rates of subsystem E and D respectively from operative to reduced states /~(x~) i = 1, 2, 3, 4 general repair rate of subsystem A, B, D and E respectively 0 operative state 3 reduced state due to failure of units in subsystem D 4 reduced state due to failure of units in subsystem E 7 reduced state due to simultaneous failure of units in subsystems D and E. Po(t) probability that the system is working to full capacity at time t; 0 is replaced by 3, 4, 7 for respective reduced states Pi(t) probability that the system is in the ith failed state at time t The limit of integration is from 0 to ~ . The transition diagram of the sytsem is shown in Fig. 1.
63 63 4 ] "Ji-'~"'-"t- Z 0~i'[-~3(X3) P3(X3' t) ~t OX3 i - 1 3
=22Po(t) + t~ j P,+,2(x,,t)~,(x,) dx, "Jr-~P7(x4, t)fld(x4) dx4, 63 63 A -I-f14(X4)] + ~X3 "[-~X4 "~ i~l ~' "~ ]~3(X3)
I~t X
P7(x3, x4, t)
'f
=
~3P4(t)+ot4P3(t)
+ ,~l= P,+ s(x,, t)fl,(xi) dxi, [fit +~xi fli(xi)]Pj(xi, t)=O,
MATHEMATICAL FORMULATION when Probability considerations give the following differential equations associated with the transition diagram:
i=1
i=1,
j=
1,5,9, 13,
i=2,
j=2,6,10,14,
i=3,
j=
i=4,
j = 8, 12.
11,15
Initial and boundary conditions are
[0
~-I q'-~-'4X4 63 "l- i~o~iW,4(x4)]e4(x4, - 1
t)
P0(0) = 1,
-- ,~ Po(t) + ,Z.= t'~+4(x~, t)#~(x~) dx~, (2)
otherwise = 0,
P,(O, t) = =if Pk(xi, t) dx i = ~iPk(t), J
Transition diagram of the system
IAooE21 ~
#,¢x,~ll',
Bz~
~
B
~
//^. l
x
(3)
,
2
0 - Operative in fuLL capacity ~-Operative in reduced capacity IZ3- FoiLed state
Fig. 1. Transition state diagram.
~,¢x,II Is,
)
(4)
(5)
A
refining system in the sugar industry
1027
when k=O,
i= 1,2,3,
j=l
k=3,
i=l,
j= i + 12,
k=4,
i=l,2,4,
j=i+4,
k=7,
i= 1, 2, 3, 4
j=i+8,
2, 3,
Fl(xd=
i ai+E3W, i-l
P,(O, 0, t) = %P,(r)
+ v,(t),
F2(x4)= i ai+ B4(x4),
P,(O, 1) =&p&h
i-l
P4(0, t) = 1, p,(t).
F
Solutions of the differential equations (l)-(S) P,(xr,-~,f)=exp
=
5
(a, +
A,),
i=l
are
1 x[M,(t)S,(x,,x,)+a,P,(t -x3) +@3(t -x4)1, [S
J’dx,,t)=ew[
&&,,x,)dx,
-SF,(x,)dx,](M,(i)S,(x,)
i$,f;,s(i)= i Ipi+*(xjt c)fii(Xi)dXi, i$,.A+ 12tf)= i pi+ 12txt3
+ Apo(t - x,)19
i=l
4l +4Po(r -
p4(x4, t) =exp
- F (x )dx [s24 xl)l~
P,(t) = e-W4,(t)
Pf2W2(~d
i=l
M&)+@,(t)+
ijYfs(f) i=l
1
i
F&r,
x4) dx,
. dx,,
=
ha4
1+
[S
[
t).
A29
P1(B1+~,)+8,(8~+12)+~+~
-I 1 .
Table 1 shows that failure of the clarifier greatly affects the system reliability. Therefore a regular watch (perhaps by an unskilled worker) on the clarifier is provided to reduce the failure rate.
i=l
M2(~)SZ(C)
x4,
In a process industry, systems are required to run for a long time, so the long-run availability A (co) of the system is calculated taking d/dt+O, us t-m. Also, taking constant repair rates, we obtain
x 1 A(m)= M,(t)&(t) =&p,(f)+ ~A+,*(0+h(t) 1 exp
h.
R(r) = P,(t) + Pr(xr, r) + P(&, t) + P,(x,,
+ 11,
[
t)Bi(xi)
Reliability of the system is given by
where %W&&~x~)=
s
1 xsexp[sF~(x~)dx,].dx~, APO(t)+
PARTICULAR CASE
lb+401 i=l
If by proper arrangement of skilled workers and maintenance planning for subsystems A, B and D, the
Table
1. Effect of failure
rate of filter, heating ~2=o.1,/3,=o.2,/?4=o.1)
and clarifier
(taking
fi, = 0.4,
Availability aI
0.0
0.005
0.01
aA
a, = 0.0
a2 = 0.025
a, = 0.05
M,= 0.075
a,=O.l
0.0 0.01 0.02
0.9707 0.9622 0.9402
0.7811 0.7756 0.7613
0.6535 0.6496 0.6396
0.5617 0.5589 0.5514
0.4926 0.4904 0.4846
0.0 0.01 1 0.02
0.9590 0.9507 0.9293
0.7736 0.7682 0.7541
0.6482 0.6444 0.6345
0.5578 0.5550 0.5476
0.4895 0.4874 0.4817
0.0 0.01 0.02
0.9477 0.9396 0.9185
0.7661 0.7608 0.7471
0.6430 0.6393 0.6295
0.5539 0.5512 0.5439
0.4866 0.4844 0.4788
1028
DINESH KuM~t et al.
failure rates are reduced to the minimum possible, i.e. ctt = ~t2 = ~t3 = 0, 22 "" 0, then, for constant repair rate, equations (1)-(5) reduce to
f = ½{(d + f ) - x/[(d + f ) 2 _ 4df ]}, b + c = 21+ ~4 + 2134,
bc = ,l, (~, +/~,) +/~42, ( d + 21)Po(t) = fl4P4(t),
(6)
~ +f14+ e4 P 4 ( t ) = 2 1 P o ( t ) + f l 4 P s ( t ) ,
(7)
d +f=
(~4 + 2fl4),
df = ~ L al = ~4 + 2fl4 + 21,
(d+
f l 4 ) P s ( t ) = %P4(t),
(8)
a0 = f14().l + ~4). COST ANALYSIS
with initial conditions P0(0) = 1,
otherwise = 0.
We consider that the service facility is always available and it remains busy for time t during the interval (0, t]. Let c~ and c2 be the revenue per unit time and ordinary service per unit time respectively. Let c3 and c4 be the cost for skilled worker and maintenance planning (including scheduled maintenance) respectively. The profit function for interval (0, t] is given by
Equations (6)-(8) give P,(t)=exp(-fl4t)[a,
fP4(t)exp(fld)dt],
II(t) = cl
f0
R ( t ) dt - c2t - (c3 + c4).
(10)
F o r optimum profit we have dH(t)/dt = 0, which gives 1
R(t) -- P0(t) + P,(t) + es(t).
(9)
Solving these integral equations from (9)
[2], we obtain
where a0
ZI "~ bcdf'
Z 3= z,=
b2-alb
+a o
b(c - b ) ( d - b ) ( f
-b)'
c 2 - a~ c + a o c(b - c)(d - c ) ( f - c) ' D2-alD D(3 - O ) ( c
+a o
-O)(f
\c: - cl zl/
(11)
d2H(t)/dt 2 is positive for t given by (11).
R ( t ) = Z l - Z 2 e -b' - Z3 e -c' - Z4 e -dr - Z5 e-:',
z2=
t =
(Z2Z3Z4Zs~b+c+d+f. log - -
-O)'
f 2 _ a l f + ao Z, = f ( b - f ) ( c - f ) ( D - f ) ' b = ½{(b + c) + x/[(b + c) 2 - 4bc]}, c = ½{(b + c) - ,f[(b + c) 2 - 4bc]},
d = ½{(d + f ) + x/[(d + f ) 2 _ 4df 1},
CONCLUSION Equation (10) gives the profit for the duration (0, t], if there is no failure in any intermediate equipment. Also, t given by (11) indicates the time for the minimum run which gives a profit, i.e. if the running period is less than t given by (11) then there will be a loss. Therefore, a profit is made only when the system is run for a time more than t [given by equation (11)]. Acknowledgement--The first author is grateful to the University Grants Commission, New Delhi, for financial support. REFERENCES
1. Dinesh Kumar, Jai Singh and I. P. Singh, Availability analysis of the feeding system in sugar industry, Microelectron. Reliab. 28(6), 867-871 (1988). 2. Vito Volterra, Theory of Functions and of Integral and Integro-differential Equations. Dover, New York (1959).
0026-2714/9053.00+ .00 © 1990PergamonPress plc
DESIGN A N D COST ANALYSIS OF A REFINING SYSTEM IN THE SUGAR INDUSTRY DINESH KUMAR,* JAI SINGHt *Mechanical Engineering Department and ?Mathematics Department, Regional Engineering College, Kurukshetra--132119 (Haryana), India and P. C. PANDEY Mechanical and Industrial Engineering Department, University of Roorkee, Roorkce (U.P.), India
(Received for publication 17 November 1989) Abstract--The role of reliability technology in the sugar industry is discussed. The industry uses various systems, viz. feeding, crushing-cum-filtering,heating-cum-sulphonationand cleaning, cooking, crystallization, separation-cure-cooling and collection for storage. We consider only the refining system, the main functionary part of the sugar industry, which consists of four subsystems A, B, D and E. The failure rate for each subsystem is taken to be constant, whereas the repair rates are variable. The analysis is carried out using a supplementary variable technique. A particular case and a profit-analysis are discussed.
INTRODUCTION In a process industry, to obtain maximum output it is necessary to run each of the units in good condition, i.e. each piece of equipment of the unit should run failure free. For this purpose, a repair facility is provided, depending upon the cost involved and the resources available. Reliability analysis helps in coordinating the failure behaviour of each piece of equipment and the repair possibility within a stipulated time with m i n i m u m loss in production. In the sugar industry (a process industry), we may use reliability technology successfully to give maximum production and to maintain quality. In the sugar industry, the juice from a feeding system [1] containing small pieces of bagasse (i.e. refuse after extracting juice from cane sugar) and m u d is subjected to filtration. The filter consists of units in series to ensure the complete removal of bagasse from juice. The bagasse-free juice is diluted with water to increase its fluidity and is sent to the heating unit. This unit is similar to a heat exchanger to exploit the maximum a m o u n t of heat produced in the system. Steam at a high temperature warms the juice and, being cooled in the process, is sent back to the boiler. The juice remains in the heater for a certain period to achieve a given pH value. Heated juice with the required pH value is sent to the sulphonation unit, where sulphur dioxide is passed through it to remove the mud. The juice is again passed through the heater. This heated juice is then sent to the clarifier where the remaining cane mud is removed by settling. The clarifier has a vibration mechanism. The heated mixture is stored in the vibrating cavity of the clarifier, and the clear juice
which separates at the upper surface is then collected in the tank. The unit has a filter with small holes which are frequently blocked by cane mud, so regular washing is provided. THE SYSTEM The system has four subsystems, A, B, D and E, as follows: (1) The filter (A) consists of two units in series. Failure of any one causes complete failure of the system. (2) The ratifier (B) consists of n units in series. Failure of any one causes complete failure of the system. (3) The sulphonation plant (D) consists of l units in parallel. Failure of units reduces the capacity of the system and hence loss in production. Complete failure occurs only when all units fail. (4) The heating plant (E) consists of m units in parallel. Failure of units reduces the capacity of the system. Complete failure occurs only when all units fail. ASSUMPTIONS (1) Failure rates for the system are constant and statistically independent. (2) Intermittent service is possible. (3) Service includes repair and/or replacement. (4) Equipment can be replaced on failure or at the time of first service call after a specified period, whichever comes first. (5) Repair rates in each subsystem are arbitrary. (6) Each subsystem has a separate repair thcility. There is no waiting time for repair in the system. (7) A repaired unit is as good as new. NOTATION
A,B,D,E
I025
indicate that units are in full operating state; a bar over upper-case letters represents reduced state, whereas lower-case letters indicate failed state
DINESH KUMAR et al.
1026
"l i = 1, 2, 3, 4 rate for complete failure of subsystems A, B, D and E respectively 2~, 22 transition rates of subsystem E and D respectively from operative to reduced states /~(x~) i = 1, 2, 3, 4 general repair rate of subsystem A, B, D and E respectively 0 operative state 3 reduced state due to failure of units in subsystem D 4 reduced state due to failure of units in subsystem E 7 reduced state due to simultaneous failure of units in subsystems D and E. Po(t) probability that the system is working to full capacity at time t; 0 is replaced by 3, 4, 7 for respective reduced states Pi(t) probability that the system is in the ith failed state at time t The limit of integration is from 0 to ~ . The transition diagram of the sytsem is shown in Fig. 1.
63 63 4 ] "Ji-'~"'-"t- Z 0~i'[-~3(X3) P3(X3' t) ~t OX3 i - 1 3
=22Po(t) + t~ j P,+,2(x,,t)~,(x,) dx, "Jr-~P7(x4, t)fld(x4) dx4, 63 63 A -I-f14(X4)] + ~X3 "[-~X4 "~ i~l ~' "~ ]~3(X3)
I~t X
P7(x3, x4, t)
'f
=
~3P4(t)+ot4P3(t)
+ ,~l= P,+ s(x,, t)fl,(xi) dxi, [fit +~xi fli(xi)]Pj(xi, t)=O,
MATHEMATICAL FORMULATION when Probability considerations give the following differential equations associated with the transition diagram:
i=1
i=1,
j=
1,5,9, 13,
i=2,
j=2,6,10,14,
i=3,
j=
i=4,
j = 8, 12.
11,15
Initial and boundary conditions are
[0
~-I q'-~-'4X4 63 "l- i~o~iW,4(x4)]e4(x4, - 1
t)
P0(0) = 1,
-- ,~ Po(t) + ,Z.= t'~+4(x~, t)#~(x~) dx~, (2)
otherwise = 0,
P,(O, t) = =if Pk(xi, t) dx i = ~iPk(t), J
Transition diagram of the system
IAooE21 ~
#,¢x,~ll',
Bz~
~
B
~
//^. l
x
(3)
,
2
0 - Operative in fuLL capacity ~-Operative in reduced capacity IZ3- FoiLed state
Fig. 1. Transition state diagram.
~,¢x,II Is,
)
(4)
(5)
A
refining system in the sugar industry
1027
when k=O,
i= 1,2,3,
j=l
k=3,
i=l,
j= i + 12,
k=4,
i=l,2,4,
j=i+4,
k=7,
i= 1, 2, 3, 4
j=i+8,
2, 3,
Fl(xd=
i ai+E3W, i-l
P,(O, 0, t) = %P,(r)
+ v,(t),
F2(x4)= i ai+ B4(x4),
P,(O, 1) =&p&h
i-l
P4(0, t) = 1, p,(t).
F
Solutions of the differential equations (l)-(S) P,(xr,-~,f)=exp
=
5
(a, +
A,),
i=l
are
1 x[M,(t)S,(x,,x,)+a,P,(t -x3) +@3(t -x4)1, [S
J’dx,,t)=ew[
&&,,x,)dx,
-SF,(x,)dx,](M,(i)S,(x,)
i$,f;,s(i)= i Ipi+*(xjt c)fii(Xi)dXi, i$,.A+ 12tf)= i pi+ 12txt3
+ Apo(t - x,)19
i=l
4l +4Po(r -
p4(x4, t) =exp
- F (x )dx [s24 xl)l~
P,(t) = e-W4,(t)
Pf2W2(~d
i=l
M&)+@,(t)+
ijYfs(f) i=l
1
i
F&r,
x4) dx,
. dx,,
=
ha4
1+
[S
[
t).
A29
P1(B1+~,)+8,(8~+12)+~+~
-I 1 .
Table 1 shows that failure of the clarifier greatly affects the system reliability. Therefore a regular watch (perhaps by an unskilled worker) on the clarifier is provided to reduce the failure rate.
i=l
M2(~)SZ(C)
x4,
In a process industry, systems are required to run for a long time, so the long-run availability A (co) of the system is calculated taking d/dt+O, us t-m. Also, taking constant repair rates, we obtain
x 1 A(m)= M,(t)&(t) =&p,(f)+ ~A+,*(0+h(t) 1 exp
h.
R(r) = P,(t) + Pr(xr, r) + P(&, t) + P,(x,,
+ 11,
[
t)Bi(xi)
Reliability of the system is given by
where %W&&~x~)=
s
1 xsexp[sF~(x~)dx,].dx~, APO(t)+
PARTICULAR CASE
lb+401 i=l
If by proper arrangement of skilled workers and maintenance planning for subsystems A, B and D, the
Table
1. Effect of failure
rate of filter, heating ~2=o.1,/3,=o.2,/?4=o.1)
and clarifier
(taking
fi, = 0.4,
Availability aI
0.0
0.005
0.01
aA
a, = 0.0
a2 = 0.025
a, = 0.05
M,= 0.075
a,=O.l
0.0 0.01 0.02
0.9707 0.9622 0.9402
0.7811 0.7756 0.7613
0.6535 0.6496 0.6396
0.5617 0.5589 0.5514
0.4926 0.4904 0.4846
0.0 0.01 1 0.02
0.9590 0.9507 0.9293
0.7736 0.7682 0.7541
0.6482 0.6444 0.6345
0.5578 0.5550 0.5476
0.4895 0.4874 0.4817
0.0 0.01 0.02
0.9477 0.9396 0.9185
0.7661 0.7608 0.7471
0.6430 0.6393 0.6295
0.5539 0.5512 0.5439
0.4866 0.4844 0.4788
1028
DINESH KuM~t et al.
failure rates are reduced to the minimum possible, i.e. ctt = ~t2 = ~t3 = 0, 22 "" 0, then, for constant repair rate, equations (1)-(5) reduce to
f = ½{(d + f ) - x/[(d + f ) 2 _ 4df ]}, b + c = 21+ ~4 + 2134,
bc = ,l, (~, +/~,) +/~42, ( d + 21)Po(t) = fl4P4(t),
(6)
~ +f14+ e4 P 4 ( t ) = 2 1 P o ( t ) + f l 4 P s ( t ) ,
(7)
d +f=
(~4 + 2fl4),
df = ~ L al = ~4 + 2fl4 + 21,
(d+
f l 4 ) P s ( t ) = %P4(t),
(8)
a0 = f14().l + ~4). COST ANALYSIS
with initial conditions P0(0) = 1,
otherwise = 0.
We consider that the service facility is always available and it remains busy for time t during the interval (0, t]. Let c~ and c2 be the revenue per unit time and ordinary service per unit time respectively. Let c3 and c4 be the cost for skilled worker and maintenance planning (including scheduled maintenance) respectively. The profit function for interval (0, t] is given by
Equations (6)-(8) give P,(t)=exp(-fl4t)[a,
fP4(t)exp(fld)dt],
II(t) = cl
f0
R ( t ) dt - c2t - (c3 + c4).
(10)
F o r optimum profit we have dH(t)/dt = 0, which gives 1
R(t) -- P0(t) + P,(t) + es(t).
(9)
Solving these integral equations from (9)
[2], we obtain
where a0
ZI "~ bcdf'
Z 3= z,=
b2-alb
+a o
b(c - b ) ( d - b ) ( f
-b)'
c 2 - a~ c + a o c(b - c)(d - c ) ( f - c) ' D2-alD D(3 - O ) ( c
+a o
-O)(f
\c: - cl zl/
(11)
d2H(t)/dt 2 is positive for t given by (11).
R ( t ) = Z l - Z 2 e -b' - Z3 e -c' - Z4 e -dr - Z5 e-:',
z2=
t =
(Z2Z3Z4Zs~b+c+d+f. log - -
-O)'
f 2 _ a l f + ao Z, = f ( b - f ) ( c - f ) ( D - f ) ' b = ½{(b + c) + x/[(b + c) 2 - 4bc]}, c = ½{(b + c) - ,f[(b + c) 2 - 4bc]},
d = ½{(d + f ) + x/[(d + f ) 2 _ 4df 1},
CONCLUSION Equation (10) gives the profit for the duration (0, t], if there is no failure in any intermediate equipment. Also, t given by (11) indicates the time for the minimum run which gives a profit, i.e. if the running period is less than t given by (11) then there will be a loss. Therefore, a profit is made only when the system is run for a time more than t [given by equation (11)]. Acknowledgement--The first author is grateful to the University Grants Commission, New Delhi, for financial support. REFERENCES
1. Dinesh Kumar, Jai Singh and I. P. Singh, Availability analysis of the feeding system in sugar industry, Microelectron. Reliab. 28(6), 867-871 (1988). 2. Vito Volterra, Theory of Functions and of Integral and Integro-differential Equations. Dover, New York (1959).