Stochastic Modelling and Reliability Analysis of a RO Membrane System Used in Water Purification System with Patience – Time for Repair

Stochastic Modelling and Reliability Analysis of a RO Membrane System Used in Water Purification System with Patience – Time for Repair

Abstract – This study presents stochastic model for analysis of real industrial system model of a RO Membrane (ROM) used in water purification system. The system consists of a single unit of RO membrane which plays important role in water purification system. In this study, two repairmans regular and expert repairman is available. If regular repairman is able to repair the failed unit within fixed amount of time known as patience-Time then it is ok otherwise the expert repairman will be called. The failure time distributions fallows negative exponential while repair time distributions are general. By using semi-Markov processes and regenerative point technique we have obtained measures of system effectiveness. The relevant datas/information gathered from industry is used for study, Graph have been plotted for economical analysis of system.

1. INTRODUCTION

   Reliability models for system have widely been analyzed by a number of authors under various assumptions, including Singh. S. K.et al (1991) analyzed cost benefit analysis of a 2-unit priority standby system with patience time for repair, Tuteja, R. K. et al (1994) discussed A system with pre inspection and two type of repairman, Chander, S. (2005) discussed Reliability model with priority for inspection and repair with arrival time of server, M. N. Gopalan et al (1985) analyzed cost benefit analysis of one- server two-unit cold standby system with repair and preventive maintenance.

   Most of these studies are not based on the real datas. However very few researchers considered the real existing system models like Singh, J.and Mahajan, P. (1999) analyzed Reliability of Utensils manufacturing plant- A case study. Guines, M. and Devici,I.(2002) discussed Reliability of service system and application in student office.Gupta,R.et al(2007) discussed benefit analysis of distillery plant system Studied some reliability models in real data of failure and repair rates in such systems. In present paper we studied performances of RO Membrane system used in water purification system model. The RO Membrane is a very important part of water purification system.

   A ROM is thin film Composite (TFC) membrane with 0.0001 micron pore size, in which the water is passed under high pressure through TFC. The output of membrane is purified water, with reduced Total Dissolved Solids TDS free from micro- organisms like bacteria, virus protozoa and

   cysts, hardness, pesticides, and heavy metals like arsenic lead and mercury.

   In view of above advantages of RO membrane in water purification industry its failure in case is not tolerable. These RO membranes may fail in following some main reasons:

   (a) Failure of electrical motor pump.

   1. Failure due to deposition of salt, dissolved solids etc

   2. Failure due to non supply of water in the membrane.

      Keeping the above view, introducing concept of reliability modelling, a one unit RO membrane system has been analyzed in the present paper in which the policy of preventive maintenance is applied after continuous working for a random amount of time to make the system more reliable. In the system two repair facilities are considered known as regular and expert. If regular repairman is able to repair the failed unit within the fixed amount of time known as patience time then it is O.K. otherwise expert repairman will be called. System is analyzed using semi- Markov process and regenerative techniques we obtain many system measures such as MTSF, availability, mean down time expected profit etc. Graphs have been plotted for economic analysis of system.

2. MODEL

   1. The system consists of a single unit R O Membrane (ROM), which is operable initially.

   2. RO membrane can fail due to following three reasons.

      1. Failure of electric motor pump.

      2. Failure due to deposition of salt, heavy metals etc.

      3. Failure due to non supply of water in the membrane

   3. The probabilities that a membrane will fail due to reasons (a), (b) and, (c) are fixed.

   4. The policy of PM is applied after continue working for random amount of time to make the system more reliable

      .In this the system becomes down (not Failed) and the complete unit inspected, flushing, servicing etc, are applied.

   5. There are two repair facilities are considered known as regular and expert. Whenever a RO membrane fails with

      any of the reasons, the failed unit is sent for repair by regular repair facility. If regular repairman is able to repair the failed unit within the fixed amount of time known as Patience time then it is O.K. otherwise expert repairman will be called. Once the expert repairman enters, it will complete all the jobs related to the system. The PM will be completed by the regular repair facility only.

   6. The distribution of time to failure of a working ROM is negative exponential while completing PM and repair of failed ROM are in general.

      1. Notations

         E Set of regenerative state

         constant failure rate of operative ROM

         constant rate of applying PM policy

         g(.)/G(.) pdf and cdf of time to complete

         PM

         p(.)/p(.)/p(.) pdf of time to complete repair

         of ROM failed due to reason of (a), (b), (c) respectively.

         H1(.)/H2(.)/H3(.) cdf of time to complete repair

         of ROM failed due to reason of (a), (b), (c) respectively

         f(.)/F(.) pdf and cdf of time to completing

         patience-time for regular repairman

         k(.)/K(.) pdf and cdf of time to complete repair of a ROM by expert repairman

         p/ q/ r probability that operative ROM will fail due to reason of (a), (b), (c) respectively

         m1 mean patience-time =

         m2 mean repair time of ROM

         by expert repairman =

         $/© Symbol for Stieltjes convolution/ Laplace convolution

         ~/* Symbol for Laplace Stieltjes Transform (LST)/ Laplace Transform

         The possible transition

      2. Symbol for States of the system

      M0 Normal unit of ROM under operative Mpm Normal unit of ROM under preventive

      maintenance

      F(a) ROM under regular repairman failed due to reason (a)

      F(b) ROM under regular repairman failed due to reason (b)

      F(c) ROM under regular repairman failed due to reason (c)

      Fex Failed ROM under expert repairman Up state S0 = (M0), Down state S1 = (Mpm) Failed state S2 = (F (a) or), S3 = (F (b) or),

      S4 = (F(c) or), S5 =( Fex),

      Fig-1. State Transition Diagram

3. TRANSITION PROBABILITIES AND MEAN SOJOURN TIME

   Simple probabilistic consideration yield the following equations for the non zero elements

   = (1)

   ,

   , ,

   ,

   ,

   , (2)

   From the above probabilities the following relation can be easily verified as

   ,

   (3)

   If is the mean sojourn time in , then mean sojourn time in state, in given by

   (4)

   Where

   T is the time of stay in state by the system.

   Therefore, the mean sojourn time for various states is as follows

   = ,

   =

   ,

   ,

   = (5)

   The unconditional mean time taken by the system to transit to any regeerative state when it is counted from epoch of entrance into is

   Thus

   = (6)

   = = , ,

   ,

   (18)

   = (19)

   6. BUSY PERIOD ANALYSIS

   6.1 Busy Period For Regular Repair Facility

   Using the probabilistic argument, we have the following recursive relation for

4. MEAN TIME TO SYSTEM FAILURE (MTSF)

To investigate the distributions function of the time to system failure with starting state S0, the failed states are taken to be absorbing. Using the arguments for a regenerative process, we obtain the following relation for

$

$ (8)

Taking LST of relation (8) and solving for by

Where (21)

(20)

omitting the argument s for brevity, we get

Where

(9)

Taking Laplace-Transform of above equation (20) and solving ) and using this, we get steady-state, the function of time for which the regular repair facility is busy in repair is given by

= (10)

= (11)

Taking the limit s0 in equation (9), one , which implies that is proper distribution function. Therefore, mean time to system failure when the initial state is , is

= = (12)

= = (22)

We have

(23)

6.2 Busy period for Expert Repair Facility

Using the probabilistic argument, we have the following recursive relation for

Where

(13)

(14)

5. AVAILABILITY ANALYSIS

Using the probabilistic arguments, we have the following recursive relations

(24)

Taking Laplace-Transform in above equations (24) and solving for and using this, we get steady-state, the function of time for which the regular repair facility is busy in repair is given by

= = (25)

Where

(15)

Where

(26)

Where

(16) Taking

= (27)

Laplace-Transform of above equation (15) and solving for and using this, we get steady-state availability of system as:

= = (17)

Where

7. EXPECTED DOWN TIME OF THE SYSTEM

Let be the probability that the system is down under PM by regular repair facility at time t. Thus following recursive relation can be obtained as:

Fig.2 Graph between MTSF and Failure rate

(28)

Taking Laplace-Transform in above equations (28) and solving and using this, we get steady-state, the function of time for which the system is down is given by

= = (29)

Where

= with (30)

8. RESULTS AND DISCUSSION

The expected profit per unit time incurred to the system is given by:

, (31)

Where

– Revenue per unit up time when system works in full capacity

– Cost per unit time for which regular repairman is busy for repairing

2- Cost per unit time for which expert repairman is busy for

repairing

3- Cost per unit up time when system is down

4- Payment per unit time made to repairmans

Particular case

Consider

,

(32)

Using the values estimated from the data collected i.e.

,

(33)

The mean time to system failure (MTSF) of the system, which decreases with the increase of failure rates of the unit shown in Fig.2. The behaviour of availability shown in Fig.3 The behaviour of the profit analysis shown in fig.4, the figure indicates that profit of the system goes on decreasing the increase of failure rates. Hence on the basis of the results obtained for a particular case, it is concluded that the system can be made more reliable and profitable when increasing the repair rates, increasing the preventive maintenance rate and revenue per unit up time increases.

6

5

MTSF

4

3

2

1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FAILURE RATE

Fig.3 Graph between Availability and failure rate

0.8

AVAILABILITY

0.7

0.6

0.5

0.4

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FAILURE RATE

Fig.4 Graph between Profit and failure rate

2500

2000

PROFIT

1500

1000

500

0

-500

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FAILURE RATE

ACKNOWLEDGEMENT:

Authors are grateful to the Department of Mathematics, Kalyan P.G. College Bhilai C.G. for providing research centre for our research work.

9. REFERENCES

1. Singh S.K., Singh R.P. (1999), Cost benefit analysis of 2-unit priority standby system with Patience-Time for Repair. IEEE Transaction on Reliab.Vo.-40, No,-1 pp 11-14, April 1999.

2. Tuteja, R.K. and Malik, S.C. (1994), A system with pre- inspection and two type of repairman, Microelectron. Reliab., vol.-32(3),pp 373-377,1994.

3. Chender, S. (2005), Reliability models with priority for inspection and repair with arrival time for server, Pure and applied Mathematika Science. Vol. LXI No. 1-2 pp 22, 2005.

4. M.N. Gopalan and H.E. Nagarwala (1985), Cost benefit analysis of one-server two-unit cold standby system with repair and P.M, Microelectron. Reliab. Vol.-25(2), pp 267-26, 1985.

5. Singh, J. Mahajan,P.(1999), Reliability of Utensils Manufacturing plant- A case study, Opsearch vol.36(3) pp 260-2691999.

6. G ines, M. and Deveci, I.(2002) Reliability of service system application in student office, Int. J. of Quality and reliability management vol.19 pp 206-211,2002.

   Gupta, R. and Kumar, K. (2007), Cost benefit analysis of Distillery plant system, Int. J. Agricult. Stat. Science. Vol.3 (2), pp 541-554. 2007.

7. Singh, J.Kumar,K. Sharma, A.(2008), Availability Evaluation of an Automobile system, Journal of Mathematical and system Science

.Vol.4 (2), pp95-100,2008.