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Performance evaluation of process plant using markov model
1Vikrant Aggarwal & 2Atul Goyal
1 Research Scholar, PTU 2 Deptt. of Mechanical Engineering, LLRIET, Moga Email: 1[email protected] 2[email protected]
ABSTRACT: This paper deals with the performance analysis of rusk making system using markovian approach.
The plant is divided into five sections like proofing unit, baking unit-I, slicing and traying unit, baking-II unit and packing unit. In the present work rusk making system has been taken for performance analysis. Failure and repair rates of these subsystems are assumed to be constant and exponentially distributed. A mathematical model pertaining to the real environment of rusk making system has been developed using Markov birth-death process. The differential equations have been derived on the basis of probabilistic approach using transition diagram. These equations are solved using normalizing conditions and recursive method to derive out the steady state availability expression of the system i.e. system’s performance criterion. The results give system availability for different combinations of failures and repair rates for various subsystems
I. INTRODUCTION
Reliability engineering had gained its importance in recent years due to the results it had shown. Reliability, in general can be defined as the probability of a system/
device performing its anticipated purpose adequately for the intended period of time under the given operating conditions. Effective maintenance management is essential and critical as a way to reduce the adverse effect of equipment failures and to maximize equipment availability. The increase in equipment availability means higher productivity and thus higher profitability provided that the maintenance optimization does include the cost factor. The optimum availability analysis is desirable for long working duration with good performance level of the systems in the industries to reducethe production cost and increase the productivity in this cut throat competition era. So, the reliability engineering is an important tool to compute and improve the systems performance which is widely used now days.Effective maintenance would certainly results in higher availability of the component and higher profits.
Because of the large number of reliability techniques, their expense, and the varying degrees of reliability required for different situations, most projects develop a reliability program plan to specify the reliability tasks that will be performed for that specific system. Recent studies by the researchers in the field of reliability/availability /maintainability [1]-[3] proposed
maintenance. An investigation [4] for the possible failure modes in the case of a helicopter had been made and recommendations were made for useful corrective measures. Research [5] had beencarried out reliability Analysis in nuclear mechanical systems. In [6]
stochastic analysis of a repairable system with three dissimilar units and two repair facilities was introduced.
The explicit expressions of the state probabilities of the system and then the explicit expressions were obtain. In [7] reliability characteristic of cold-standby redundant system was introduced considering two units. In [8]
some reliability parameters of a three state repairable system with environmental failure were evaluated.
Using [9] the Markovian approach, on three state system, the formulae for steady state availability, the frequency of failure, mean time to failure and mean duration of down was derived. A model [10] on multi- state series, parallel and series parallel and parallel- series systems with regular reliability structures, considering the components that have exponential reliability functions with different transition rates between subsets of their states was studied expressions [11] for reliability and mean time to failure of k-out-of-n systems. The importance of various analytical [12]and simulation techniques for availability modelling and effective assessment of continuous production systems with the main objective of economic optimization. The need for carrying availability analysis at the component level rather than targeting at system level had been discussed.A rusk manufacturing plant situated in Ludhiana, India is chosen for study. In this paper a subsystem of rusk making manufacturing system, which is a continuous production system is considered and the availability analysis of the complex mechanical system under preemptive resume priority repair is carried out.
Laplace transform is used for solving differential equations to obtain state probabilities. Numerical results based on the true data collected from industry are presented to illustrate the steady state behavior of the system under different plant conditions.
II. SYSTEM DESCRIPTION
In Rusk making system, the rusks are made from dough sheets. Proofing refers to the dough resting period during fermentation for desired dough height at 30-35ºC and at 85% relative humidity for 60 minutes. During
proofing. In the baking unit, the dough is baked at 240 0 C for 25 minutes on a belt conveyor and then cooled for 10 minutes. In the slicing and traying unit the baked bread is sliced according to the need and put in the trays keeping one side up. After that for 10 minutes at temperature of 300 0 C the one side is baked , in the oven itself the side is changed .Inthe packing unit the rusks are packed as per the requirement. Figure 1 represents the description of the process.
Flour Sheets
Rusk packets
Fig. 1 Process Flow Diagram
III. NOTATIONS AND ASSUMPTIONS
Notations
Proofing unit (P): One unit subjected to major failure only.
Baking unit-I (B): One unit subjected to major failure only.
Slicing & Traying (S): One unit subjected to major failure only.
Baking unit-II (K): One unit subjected to major failure only.
Packing unit (C): One unit subjected to major failure only.
λj : Indicate failure rates of P,B,S,C & K (j=1,2,3,4,5).
µj : Indicate repair rates of P,B,S,C & K (j=1,2,3,4,5).
o : Indicate Component/Sub-system is operative.
q : Indicate component/Sub-system is in queue for repair.
r : Indicate component/Sub-system is under repair.
Assumptions
a) All the subsystems are initially operating.
b) All the sub-systems are initially in good state.
c) Each unit has two states viz., good and failed.
d) It is also assumed that there is only one repair facility and priority will be given to sub-system P, B, S, C& K for repair activity.
e) Each unit is as good as new after repair.
f) The failure rates and repair rates of all units are taken constant.
g) Failure and repair events are statistically independent.
IV MATHEMATICAL ANALYSIS OF THE SYSTEM
Probability consideration gives the following first order differential-difference equations associated with the state transition diagram of the system.
(λ1+λ2+λ3+λ4+λ5)P0(t)=µ1P1(t)+µ2P2(t)+µ3P3(t+µ4P
4(t) +µ5P5(t)
+α1P0(t)=µ1P1(t)+µ2P2(t)+ µ3P3(t)+µ4P4(t) +µ5P5(t) Where α1= λ1+ λ2+ λ 3+ λ4+ λ5
+µkPk(t)=λkP0(t) Where k= 1,2,3,4,5
System working at System in failed full capacity state
Fig. 2 State Transition Diagram
Taking Laplace transform of the above equations, which gives the following equations
(1) (2) (3)
λ2
µ1 µ4
λ1
µ2 λ2
µ5 λ5
(1) P
r, B
g, S
g, C
g, K
gProofing Unit
Baking Unit-I
Slicing & Traying Unit
Baking Unit-II
Packing unit
µ
5
µ
6
Λ
6
µ
5
µ
6
Λ
6 µ3
λ
2
λ
2
λ
3(0) P0, B0, S0, C0, K0 (4)
Pg, Bg, Sg, Cr, Kg
(3) Pg, Bg, Sr, Cg, Kg
(1) Pr, Br, Sg, Cg, Kg
(5) Pg, Bg, Sg, Cg, Kr
(2) Pg, Br, Sg, Cg, Kg
λ
4(4) (5) (6) Solving Equations (2), (3), (4), (5) and (6)
Where K1 =
Where K2 =
Where K3 =
Where K4 =
Where K5 =
Put the value of P1(s), P2(s), P3(s), P4(s) and P5 (s) in equation (1) and by using the normalizing condition, we get
Availability function for A(s) system is given as A(s) =
Inversion of A(s) gives the availability function A(t) Applying steady state behavior on fourth order differential equations when
t , . We get
Similarly from equations (2), (3), (4), (5),(6) we get ; ; ; ;
Using normalizing condition
……...(A)
Steady state availability of the system is given as A ( ) =
Where is given by equation (A)
A ( ) = 0.8731 taking the values as mention in table No.1
Table 1
λ1 λ2 λ3 λ4 λ5
0.005 0.01 0.007 0.01 0.008
µ1 µ2 µ3 µ4 µ5
0.30 0.33 0.25 0.33 0.2
V. AVAILABILITY ANALYSIS
The effect of various parameters on availability is studied. If the failure and repair rates are varied, the availability is affected. This effect is shown in following tables obtained for availability analysis of rusk making system.
a)Effect of failure rate of Proofing unit on availability A ( ) :
λ2 = 0.01, λ3 =0.007, λ4 =0.01, λ5 =0.008, µ1=0.30, µ2=0.33, µ3= 0.25, µ4= 0.33, µ5= 0.2
Table 2. Steady state availability versus failure rate of proofing unit
λ1 0.005 0.010 0.015 0.02 0.025 A 0.8731 0.8606 0.8484 0.8366 0.8251 b) Effect of failure rate of Baking unit on availability A ( ) :
λ1 = 0.005, λ3 =0.007, λ4 =0.01, λ5 =0.008, µ1=0.30, µ2=0.33, µ3= 0.25, µ4= 0.33, µ5= 0.2
Table 3. Steady state availability versus failure rate of baking unit
λ2 0.01 0.02 0.03 0.04 0.05 A 0.8731 0.8506 0.8292 0.8089 0.7895 c) Effect of failure rate of slicing & traying on availability A ( ) :
λ1 = 0.005, λ2 =0.01, λ4 =0.01, λ5 =0.008, µ1=0.30, µ2=0.33, µ3= 0.25, µ4= 0.33, µ5= 0.2
Table 4. Steady state availability versus failure rate of slicing & traying unit
λ3 0.007 0.014 0.021 0.028 0.035 A 0.8731 0.8523 0.8324 0.8134 0.7953
d) Effect of failure rate of Baking unit-II on availability A ( ) :
λ1 = 0.005, λ2 =0.01, λ3 =0.007, λ5 =0.008, µ1=0.30, µ2=0.33, µ3= 0.25, µ4= 0.33, µ5= 0.2
Table 5. Steady state availability versus failure rate of baking-II unit
Λ4 0.01 0.02 0.03 0.04 0.05 A 0.8731 0.8506 0.8292 0.8089 0.7895 e) Effect of failure rate of packing unit on availability A ( ) :
λ1 = 0.005, λ2 =0.01, λ3 =0.007, λ4 =0.001, µ1=0.30, µ2=0.33, µ3= 0.25, µ4= 0.33, µ5= 0.2
Table 6. Steady state availability versus failure rate of packing unit
λ5 0.008 0.016 0.024 0.032 0.04 A 0.8731 0.8436 0.8161 0.7903 0.7661 f) Effect of repair rate of proofing unit on availability A ( ):
λ1=0.005,λ2 =0.01, λ3 =0.007, λ4 =0.001, λ5 =0.008, µ2=0.33, µ3= 0.25, µ4= 0.33, µ5= 0.2
Table 7. Steady state availability versus repair rate of proofing unit
µ1 0.3 0.6 0.9 1.2 1.5
A 0.8731 0.8795 0.8817 0.8827 0.8834 (g) Effect of repair rate of baking unit on availability A ( ):
λ1=0.005, λ2 =0.01, λ3 =0.007,λ4 =0.001,λ5 =0.008, µ1=0.30, µ3= 0.25, µ4= 0.33, µ5= 0.2
Table 8. Steady state availability versus repair rate of baking unit
µ2 0.33 0.66 0.99 1.32 1.65 A 0.8731 0.8848 0.8888 0.8908 0.8920 (h) Effect of repair rate of Slicing & Traying unit on availability A ( ):
λ1=0.005, λ2 =0.01, λ3 =0.007, λ4 =0.001, λ5=0.008, µ1=0.30, µ2= 0.33, µ4= 0.33, µ5= 0.2.
Table 9. Steady state availability versus repair rate of Slicing & Traying unit
µ3 0.25 0.5 0.75 1.00 1.25 A 0.8731 0.8839 0.8876 0.8894 0.8905
i) Effect of repair rate of baking unit-II on availability A ( ):
λ1 = 0.005, λ2 =0.01, λ3 =0.007, λ4 =0.001, λ5 =0.008, µ1=0.30, µ2= 0.33, µ3= 0.25, µ5= 0.2
Table10. Steady state availability versus repair rate of baking unit-II
µ4 0.33 0.66 0.99 1.32 1.65
A 0.8731 0.8848 0.8888 0.8908 0.8920 j) Effect of repair rate of packing unit on availability A ( ) :
λ1 = 0.005, λ2 =0.01, λ3 =0.007, λ4 =0.001, λ5 =0.008, µ1=0.30, µ2= 0.33, µ3= 0.25, µ= 0.33
Table 11. Steady state availability versus repair rate of packing unit
µ5 0.2 0.4 0.6 0.8 1.0
A 0.8731 0.8886 0.8939 0.8966 0.8982
VI. RESULTS AND DISCUSSION
The unit availability has been excellent, mainly because of the low failure rate, supported by the state of the art repair facilities A Markov model is presented to predict operational availability of rusk making unit of process industry .It is observed that for some known values of failure/repair rates of proofing unit, baking unit, slicing and traying unit and packing unit, plant availability reduces by approximately 11%. On the contrary increase in the repair rate increase the availability of the system.
With increase in repair rate there had been increase of 1.03%, 1.89%, 1.74%, 1.89% and 2.51% respectively in proofing unit, baking unit, slicing and traying unit, baking-II unit and packing unit respectively.So findings of this paper will be highly beneficial to the plant manager for properly executing the maintenance tasks and hence enhance the performance of plant.
VII. REFERENCES
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