Reliability Analysis of a Seven Unit Desalination Plant with Shutdown during Winter Season and Repair/Maintenance on FCFS Basis

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1 International Journal of Perforability Engineering Vol. 9, No. 5, Septeber 2013, pp RAMS Consultants Printed in India Reliability Analysis of a Seven Unit Desalination Plant with Shutdown during Winter Season and Repair/Maintenance on FCFS Basis VIZS. M. RIZWAN *1, N. PADMAVATHI 2, ANITA PAL 3, and G. TANEJA 4 1, 2 Departent of Matheatics & Statistics, Caledonian College of Engineering, OMAN 3 Departent of Matheatics, National Institute of Technology, Durgapur, INDIA 4 Departent of Matheatics, M.D.University, Rohtak, INDIA (Received on February 21, 2013, revised on June 09, 2013) Abstract: In any desalination plants, ulti stage flash desalination process is norally used for sea water purification and the reliability analysis of such a coplex syste with standby support echanis is of great iportance to avoid big loses. Thus, the ai of this paper is to present a reliability analysis of evaporators of a desalination plant. The desalination plant operates round the clock and during the noral operation; six of the seven evaporators are in operation for water production while one evaporator is always under scheduled aintenance and used as standby. The coplete plant is shut down for about one onth during winter season for annual aintenance. The water supply during shutdown period is aintained through ground water and storage syste. Any ajor failure or annual aintenance brings the evaporator/plant to a coplete halt and the plant goes under forced outage state. For the present analysis, seven years failure data has been extracted fro operations and aintenance reports of a desalination plant in Oan. Various easures of the plant effectiveness have been obtained probabilistically. Sei- Markov processes and regenerative point techniques are used in the entire analysis. Keywords: Desalination plant, aintenance, failures, shutdown, sei-markov, regenerative process. 1. Introduction Desalination is a water treatent process that reoves the salt fro sea water or brackish water. It is the only option in arid regions, since the rainfall is arginal. In any desalination plants, ulti stage flash desalination process is norally used for water purification which is very expensive and involves sophisticated systes. Since, desalination plants are designed to fulfil the requireent of water supply for a larger sector in arid regions, they are norally kept in continuous production ode especially during suer except for eergency/forced/planned outages. It is therefore; very iportant that the efficiency and reliability of such a coplex syste is aintained in order to avoid big loses. Many researchers have spent a great deal of efforts in analysing industrial systes to achieve the reliability results that are useful for effective equipent/plant aintenance. Bhupender and Taneja [1] analysed a PLC hot standby syste based on aster-slave concept and two types of repair facilities, and any such analyses could be seen in the references therein. Mathew et al. [2] have recently presented an analysis of an identical two-unit parallel CC plant syste, using the real data. For general reference, a book authored by Kuo and Zuo [3] ay be consulted. Thus, the ethodology for syste analysis under various failure and repair assuptions has been widely presented in the literature and the novelty of this work lies in its case study. Desalination being coonly *Corresponding author s eail: rizwan@caledonian.edu.o 523

2 524 S. M Rizwan, N. Padavathi, Anita Pal and G. Taneja used industrial processes in arid regions; the nuerical results of various reliability indices could be helpful in understanding the significance of these failures/aintenances on plant availability and assess the ipact of these failures on the overall profitability of the plant. Thus, the paper is an attept to present a case analysis using the failure data of seven years fro operations and aintenance reports of the desalination plant in Oan. Coponent failure, aintenance, and plant shutdown rates, and various aintenance costs involved are estiated fro the data. The situation considered in this analysis is the sae as that of the plant. The desalination plant operates round the clock and during the noral operation; six of the seven evaporators are in operation for water production while one evaporator is always under scheduled aintenance and used as standby evaporator. This ensures the continuous water production with iniu possible failures of the evaporators. The coplete plant is shut down for about a onth during winter season because of the low consuption of water for annual aintenance; the water supply during this period is aintained through ground water and storage syste. Any ajor failure or annual aintenance brings the evaporator/plant to a coplete halt and goes under forced outage state. The unit/evaporator is then repaired or aintained on FCFS basis. Using the data, following values of rates and various costs are estiated: Estiated rate of failure of any coponent of the plant (λ) = per hour Estiated rate of Maintenance (γ) = Estiated rate of shutting down of the plant (γ 1 ) = per hour Estiated rate of winter to suer change (β 1 ) = per hour Estiated rate of suer to winter change (β 2 ) = per hour Estiated rate of recovery after shut down during winter (γ 2 ) = per hour Estiated value of repair rate (α) = per hour Estiated costs: C o = OMR 596, C 1 = OMR 2.054, C 2 = OMR 0.063, C 3 = OMR 0.673, C 4 = OMR The plant is analysed by using sei-markov processes, and regenerative point techniques; and the easures of plant effectiveness in ters of reliability indices such as; the ean tie to shut down the syste, ean tie to failure of one evaporator, evaporators availability, the expected busy period for aintenance, busy period for repair, expected busy period during shut down and the expected nuber of repairs need replaceent are estiated nuerically. The ipact of failures and outages on the plant availability could be assessed through these indices, and the overall profitability of the plant. FCFS OMR O U s U wb U wa F F rs F rwb Acrony Notation First Coe First Served Oani Riyal Operative state of evaporator Under aintenance during suer Under aintenance during winter before service Under aintenance during winter after service Failed state of one of the evaporator Failed unit is under repair during suer Failed unit is under repair during winter before service

3 Reliability Analysis of a Seven Unit Desalination Plant with Shutdown During 525 Winter Season and Repair/Maintenance on FCFS Basis F rwa F RS F RWB F RWA β 1 β 2 λ γ γ 1 γ 2 α p ij q ij (t), Q ij (t) g(t), G(t) g (t), G (t) C 0 C 1 C 2 C 3 C 4 A 0 B 0 M B 0 R B 0 S R 0 Failed unit is under repair during winter after service Repair of failed unit under repair during suer, is continued fro the previous state Repair of failed unit under repair during winter before service, is continued fro the previous state Repair of failed unit under repair during winter after service, is continued fro the previous state Rate of suer to winter change Rate of winter to suer change Rate of failure of any coponent of the evaporator Rate of aintenance Rate of shutting down Rate of recovery after shut down during winter Rate of repair Sybol for Laplace Convolution Transition probability fro regenerative states i to a regenerative state j p.d.f. and c.d.f. of first passage tie fro a regenerative state i to a regenerative state j in (0, t] p.d.f. and c.d.f. of repair rate p.d.f. and c.d.f. of aintenance rate Revenue per unit up tie Cost per unit up tie for which the repairan is busy for aintenance Cost per unit up tie for which the repairan is busy for repair Cost per unit up tie for which the repairan is busy during shutdown Cost per unit repair require replaceent Steady state availability of the syste Expected busy period of the repairan for aintenance Expected busy period of the repairan for repair Expected busy period of the repairan during shutdown Expected nuber of repairs require replaceent (All costs in exaples have been reckoned in Oani Riyal) 2. Model Description and Assuptions 1. There are seven evaporators in the desalination plant; of which 6 operate at any given tie and one evaporator is always under scheduled aintenance. 2. The plant goes into shutdown state for annual aintenance during winter season for one onth. 3. The various states of the evaporator are categorized under suer (states 0, 2, 5, and 8), winter (states 1, 4, 7, 10), coplete shutdown for overhaul/ajor service (state 3), and after ajor service (states 6, 9, 11, 12). 4. The failed units are repaired or aintained on FCFS basis. 5. If a unit of an evaporator is failed in one season, it gets repaired in that season only. 6. Maintenance of no evaporator is done if the repair of soe other evaporator is going on. 7. The cost of plant aintenance is high during suer due to ore requireent of water than winter where the requireent goes down. 8. Not ore than two evaporators fail at a tie.

4 526 S. M Rizwan, N. Padavathi, Anita Pal and G. Taneja 9. All failure ties are assued to have exponential distribution whereas the other ties have general distributions. 3. Transition Probabilities and Mean Sojourn Tie Figure 1: Transition states of the syste A state transition diagra showing the possible states of transition of the plant is shown in Figure 1. The epochs of entry into states 0, 1, 2, 3, 4, 5, 6, 7 and 11 are regeneration points and hence these states are regenerative states. States 8, 10 and 12 are non-regenerative states. The transition probabilities are given by: dq 00 = γe (6λ+ β 1+γ )t dt, dq 01 = β 1 e (6λ+ β 1 )t (t)dt, G dq 02 = 6λe (6λ+ β 1 )t G (t)dt; dq 11 = γe (6λ+ γ 1+γ )t dt, dq 13 = γ 1 e (6λ+ γ 1 )t G (t)dt, dq 14 = 6λe (6λ+ γ 1 )t G (t)dt; dq 24 = β 1 e β 1t G (t)dt, dq 25 = e β 1t g (t)dt, dq 36 = γ 2 e γ 2t dt; dq 43 = γ 1 e γ 1t G (t)dt, dq 47 = e γ 1t g (t)dt; dq 50 = e (6λ+ β 1 )t g(t)dt, dq 57 = β 1 e (6λ+ β 1 )t G (t)dt, (8) dq 55 6λe (6λ+ β 1 )t e β 1t g(t)dt, (8,10) dq 53 = 6λe (6λ+ β 1 )t β 1 e β 1t γ 1 e γ 1t G (t)dt (8,10) dq 57 = 6λe (6λ+ β 1 )t β 1 e β 1t e γ 1t g(t)dt; dq 60 = β 2 e (6λ+ β 2 )t dt, dq 66 = e (6λ+ β 2 )t g (t),

5 Reliability Analysis of a Seven Unit Desalination Plant with Shutdown During 527 Winter Season and Repair/Maintenance on FCFS Basis dq 69 = 6λ e (6λ+ β 2 )t G (t)dt; dq 71 = e (6λ+ γ 1 )t g(t)dt, dq 73 = γ 1 e (6λ+ γ 1 )t G (t)dt, (10) dq 73 = (6λ e 6λt γ 1 e γ 1t ) G (t)dt, (10) dq 77 = 6λe (6λ+ γ 1 )t e γ 1t g(t)dt; dq 92 = β 2 e β 2t (t)dt, G dq 9,11 = e β 2t g (t)dt; dq 11,5 = β 2 e (6λ+ β 2 )t G (t)dt, dq 11,6 = e (6λ+ β 2 )t g(t)dt, (12) dq 11,11 = 6λe (6λ+ β 2 )t e β 2t g(t)dt, (12,8) dq 11,5 = 6λe (6λ+ β 2 )t β 2 e β 2t e β1t g(t)dt. The non-zero eleent p ij can be obtained as: p ij = li s 0 q 0 ij (t) dt The ean sojourn tie ( μ i ) in the regenerative state i is defined as the tie of stay in that state before transition to any other state. If T denotes the sojourn tie in the regenerative state i, then µ i = E(T) = P(T > t); 1 µ 0 =, µ 1 (6λ+ β 1 +γ ) 1 =, µ (6λ+ γ 1 +γ ) 2 = 1, µ ( β 1 +γ ) 3 = 1, µ γ 4 = 1 2 ( γ 1 +γ ), 1 µ 5 =, µ 1 (6λ+ β 1 ) 6 =,µ 1 (6λ+ β 2 +γ) 7 =, µ (6λ+ γ 1 +α) 9 = 1, µ 1 ( β 2 +γ ) 11 = (6λ+ β 2 +α) The unconditional ean tie taken by the syste to transit for any regenerative state j when it (tie) is counted fro the epoch of entry into state i is atheatically stated as: ij = tdq ij (t) = q ij (0), ij = µ i 0 j = µ = µ = µ 2 36 = µ = µ (8) 55 + (8,10) 53 + (8,10) 57 = µ = µ (10) ( ) 77 = µ ,11 = µ 9 11,5 + 11,6 + (12) 11,11 + (12,8) 11,5 = µ 11 Regarding the failed states as absorbing states, and eploying the arguents used for regenerative processes, we obtain the recursive relations for the reliability indices; the ean tie to shut down the plant, ean tie to failure of one evaporator, availability analysis of the plant, expected busy period for aintenance, expected busy period for repair, expected busy period during shut down, and the expected nuber of repairs. By taking the Laplace or Laplace Stieltje s transfors of recursive relations, whichever is applicable; the steady-state solutions for these easures of plant effectiveness are obtained.

6 528 S. M Rizwan, N. Padavathi, Anita Pal and G. Taneja 4. Profit Analysis One of the ain objectives of the reliability analysis is to have cost-effective and profitable aintenance strategies. In order to reflect this, the overall profit of the syste could be defined; by incorporating the steady-state solutions and various costs: M R S P=C0A0 -C1B0 -C2B0 -C3B0 -C4R0 5. Particular Case: Let us assue that: g(t) = αe αt, g (t) = γe γt, g s (t) = γ 1 e γ 1t, g r (t) = γ 2 e γ 2t Using the data as suarized in section 1, the following values of easures of plant effectiveness are obtained: Mean tie to shut down the plant = hours Mean tie to failure of one evaporator = 7752 hours Availability (A 0 ) = Expected Busy period for Maintenance (B 0 M) = Expected Busy period for repair (B 0 R) = Expected Busy period during shutdown (B 0 S) = Expected nuber of repairs (R 0 ) = Profit (P) = RO 591 per unit tie. References [1]. Bhupender Parashar and Gulshan Taneja. Reliability and Profit Evaluation of a PLC Hot Standby Syste based on a Master-Slave Concept and Two Types of Repair Facilities. IEEE Transactions on Reliability 2007; 56(3): [2]. A. G. Mathew, S. M. Rizwan, M. C. Majuder, K. P. Raachandran and G. Taneja. Reliability Analysis of an Identical Two-Unit Parallel CC Plant Syste Operative with Full Installed Capacity. International Journal of Perforability Engineering March 2011, 7(2): [3]. Kuo, W. and M. J. Zuo, Optial Reliability Modeling: Principles and Applications. John Wiley & Sons, Incorp., USA; S M Rizwan is currently Professor & Head of the Departent of Matheatics and Statistics, Caledonian (University) College of Engineering, Sultanate of Oan. He obtained Ph.D. in Reliability Modeling and Analysis. He has about 20 years of teaching and research experience. His research interest is Reliability Analysis and Stochastic Processes, in which area and related ones he has published about 65 research papers. N Padavathi is working as Senior Lecturer in the departent of Matheatics and Statistics, Caledonian (University) College of Engineering, Sultanate of Oan. Her research interest is Reliability Analysis of systes in which area she has published quite a few research papers. She has about 22 years of teaching experience. Anita Pal is Assistant Professor in Matheatics at National Institute of Technology, Durgapur, India. She has about 7 years of teaching experience and a nuber of research publications to her credit with a special interest in Coputational Graph theory. Gulshan Taneja is currently Professor, Departent of Statistics, University College, MD University, India. He has published over 35 research papers in the journals of repute. Over 30 research papers have been presented in 13 national / international conferences. His area of research is Reliability Modeling and Analysis.