STOCHASTIC ANALYSIS OF AN AIR CONDITION COOLING SYSTEM MODEL

Transcription

1 Journal of Reliability and Statistical Studies; ISSN (Print) (Online) Vol. 4 Issue 1 (011) STOCHASTIC ANALYSIS OF AN AIR CONDITION COOLING SYSTEM MODEL Rakesh Gupta1 Punam Phartyal and O.K. Belwal Department of Statistics C.C.S. University Meerut UP Department of Statistics HNB Garhwal University Srinagar-Garhwal UK 1 Abstract The present study deals with the stochastic analysis of a real existing industrial system model of a central air-condition (AC) system. The system consists of three different subsystems namely- Air Blower Compressor water pump. All these subsystems are arranged in series network. Transition probabilities as well as the recurrence relations for various reliability and cost effective measures are developed. Failure time distributions of all the subsystems are taken as exponential whereas repair time distributions are general. By using regenerative point technique we have obtained various measures of system effectiveness such as Reliability MTSF Availability Busy period of repairman and Net expected profit. The results are also drawn in a particular case when repair time distributions are assumed as exponentials. Key words Reliability MTSF Availability Profit analysis regenerative point technique. 1. Introduction The reliability of hypothetical models have been analyzed widely by various authors including Gupta and Goel (1990) Gupta et al. (1994) Gupta and Chaudhary (1994). They obtained cost benefit analysis and various measures of system effectiveness by using different techniques. But sometimes the hypothetical models are not accurately reflected with the real existing systems. Few authors considered the real existing system models like Gupta and Shivakar (003) analyzed a stochastic model of cloth weaving system Gupta and Kumar (007) studied a distillery plant system and obtained profit function and reliability characteristics. Arora et al. (000) Kumar et al. (1996) also worked with real existing industrial systems with different techniques. The present study is devoted to the stochastic analysis of real existing industrial AC system model. The system is of a complex type reparable engineering system installed in majority of office buildings scientific laboratories industrial and commercial complexes etc. The AC system consists of three main subsystems that are (i) Air Blower (ii) Compressor and (iii) Water pump. These sub systems are arranged in a series configuration and the system failure occurs if any one of the subsystem fails.. System Discription The AC system consist three main subsystems that are Air blower compressor and water pump. Out of three subsystems two subsystems (compress and water-pump) have their cold standby units. Air Blower is working as single (without redundancy) as it is too much expensive. A single repairman is always available with the system to repair the failed unit and operate the cold standby unit with the help of a switching device which is always perfect and instantaneous whenever required. The service discipline of the repair man is FCFS (first come first serve).

2 96 Journal of Reliability and Statistical Studies June 011 Vol. 4 (1).1 Roll of Subsystems Air Blower (B) Air blower sucks the hot air from air-conditioning rooms and blows it into the condenser to cool down with the help of FREON gas that present inside the condenser. Compressor (C) The main work of compressor is to suck high-temperature-lowpressure (HT-LP) super heated vapour form FREON gas from air-blower to compressor and compress it into high-pressure liquid form gas with the help of pistons and pass to water condenser to cool and convert in to the form of low-temperature-high-pressure super cool compressed liquid gas. Water Pump (Wp) The main function of water pump is to pump cold water from cooling tower to water condenser in order to chill FREON gas to maintain low temperature.. Notations and States of The System b c w Constant failure rates of blower compressor and water G b ( ) G c ( ) G w ( ) pump. c.d.f. of repair time of blower compressor and water q ij pump. p.d.f. of transition time from state p ij steady state direct transition probability from state to Z i (t) i * ~ Si to S j. Si S j such that pij qij(u)du Si up to time t. Probability that system sojourns in state Mean sojourn time in state Si. symbols for Laplace and Laplace - Stieltjes transforms. To write the various states of the system we define the following symbols B o /B r /B wo Blower is operative/under C o /C s /C r /C wr /C F /C wo repair/ waiting for operation. Compressor is operative/ Wp o /Wp s /Wp r /Wp wr /Wp F /Wp wo standby/ under repair/wait for repair/ failed/waiting for operation. Water pump is operative/ standby /under repair/waiting The limits of integration are 0 to whenever not mentioned. 96

3 Stochastic Analysis of an Air Condition The which S 0 possible states of 97 the system for repair/ failed/ waiting for operation. S 0 to S 14 in are S 1 S S 4 S 7 are operative states and other are failed. Transition diagram of the system model is shown in figure Transition Probabilities and Sojourn Times (a) The direct or one step steady state transition probabilities are as follows The steady state transition probabilities can be obtained by using the results p 01 lim Q 01 (t) t c e ( b c w )u du c c b c w Where b c w Similarly b 1 G c ( ) w b p 03 p 10 G c ( ) p 15 c 1 G c () c 1 G w () p 16 p 0 G w ( ) p 7 b 1 G w () w 1 G w ( ) p 8 p 9 w 1 G c ( ) p 30 dg b (u) 1 p G c ( ) p 410 b 1 G c ( ) c 1 G c () p 411 p 4 p 51 dg b (u) 1 p G w () p 0 b 1 G w () c 1 G w () p 7 w 1 G w ( ) p 8 dg b (u) 1 dg w (u) 1 p 3 p 4 p 10 4 Similarly p p 7 1 p p (6) 11 c 1 G c () p 16

4 98 Journal of Reliability and Statistical Studies June 011 Vol. 4 (1) p w ve ( )v p 111 b w v e p 1 c w p 1 c p c ve p 14 c w ( )v ve ve G c (v)dv G c (v)dv dg w (v) p 13 b c ( )v ( )v G c (v)dv ( )v ( )v ve dg c (v) p 110 w v e ve ( )v G w (v)dv G w (v)dv ( )v G w (v)dv c 1 e ( )v dg c (v) p 4 ( )v p w 1 e dg w (v) p 7 14 ( )v w dg (v) p p (9) 1 e w 9 (1 e ( )w ) w e ( )w (7 14) dg w (w) p p 4 c w 14 () ( ) p p (4) c w (1 e ( )w ) w e ( )w dg c (w) p 1 () ( ) Hence we observe that p 01 p 0 p 03 1 p 10 p (6) p p 15 p p (9) (7 14) p 0 p p 8 p 1 p p 4 13 p 1 p p p 410 p p p p 7 13 p 7 1 p 30 p 51 p 8 p 10 4 p 11 4 p 137 p 7 1 (1-6) 3.1 Mean Sojourn Times If Ti is the sojourn time in state S i then mean sojourn time in state S i is given by 98

5 Stochastic Analysis of an Air Condition 99 P( Ti > t) dt i Therefore the mean sojourn times for various states are as follows 0 e ( b c w )u du ( b c w )u 1 b c w G c (u)du e e ( )u G w (u)du 3 G b (u)du e ( )u G c (u)du 6 G c (u)du 7 e ( )u G w (u)du 9 G w (u)du b c w b c w b c w (7-14) 4. Analysis of Results 4.1 Reliability and MTSF Let Ri (t) be the probability that the system is operative during (0 t) given that at t0 it starts from state S i E. By simple probabilistic arguments the value of R 0 (t) in terms of its Laplace transforms is given by R 0 (s) Where N 1 (s) (15) D 1 (s) N 1 (s) Z 0 1 q q Z 1 +q 14 Z 4 q 01 q q Z q 7 Z 7 q 01 q q 0 (16) D 1 (s) 1 q 1 q q 10 q 01 q 1 q 0 + q 0 q 01 q q 0 () Taking the inverse Laplace Transform of (15) one may get the reliability of the system when initially system starts from state S 0. In particular case when repair time distributions are also exponential by using the matlab software we obtained the values of reliability of the system for different values of mission time and the curves are drawn in figure-. The mean time to system failure (MTSF) can be obtained by using the well known formula E(T0 ) R 0 (t) dt Now using the results lim R 0 (s) s 0 q ij (0) p ij and 99 N 1 (0) D 1 (0) Z i 0 i we get

6 100 Journal of Reliability and Statistical Studies June 011 Vol. 4 (1) p p p D (0) 1 p p p p p N 1 (0) 0 1 p p 1 +p 14 4 p 01 p p p 0 p (18) + p p 0 01 p p 0 (19) 4. Availability Analysis A i (t) be the probability that the system is up (operative) at epoch t when initially the system starts from state S i E. Using the technique of the Laplace Let transform the value of A 0 (t) in terms of its L.T. i.e. A 0 (s) (0) Now the steady state probability that the system will be operative is given by A 0 lim s A 0 (s) N /D (1) s 0 Where N U0 0 U1 1 U U3 3 U 4 4 () D U 0 0 U 1 n 1 U n p 03 U 0 n 3 U 4 n 4 p 15 U n 5 p 110 U1 p 410 U 3 n 8 p U 1 p 411 U 3 n 9 p U p 7 13 U 4 n p U p 7 U 4 n 11 Where (3) p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1 p p p p p p p p p p p p p p p U 0 1 p (6) p 15 1 p (9) p 8 11 (7 14) (9) (7 14) p (4) 13 (4) (7 14) 4 (4) (9) U 1 p p p p p p p 01 p 0 p 1 p 01 p p 01 p 8 100

7 Stochastic Analysis of an Air Condition ) p 0 p p (7 p p p p p 13 4 U p p p p p p p p p 1 p p p 01 p p p p p p p (4) p 0 p p p p p (4) p p U p p p p p p p 1 p p p p (4) p p p p p 0 p 111 p p p p ) (4) p 0 p p p p p p (7 p ) p p p 1 p p (6) U 4 p (7 p p p p p (4) (9) p p p p p 111 p p 0 p 1 p 01 1 p p ) p 01 p p (7 p p p p 13 p 4 3 (7 14) (6) (6) ) (4) p 01 p p p p p p (7 p The expected up time of the system during (0 t) is given by up (t) t 0 A 0 (u) du (4) 4.3 Busy Period Analysis (t) and B iwp (t) be the probabilities that the system is under repair at epoch t when the system initially starts from regenerative state S i E. Using c B Let B i (t) B i elementary probabilistic arguments the value of B 0c (t) B B0 (t) and B 0Wp (t) can be c B Wp obtained in terms of their Laplace transforms i.e. B 0 (s) B 0 (s) and B 0 Now the steady state probabilities (s). B 0c B B0 and B 0wp that in the long run the probability that the repair facility will be busy in the repairing of failed compressor blower room and water pump respectively are given by So that B 0c N 3 /D B B0 N 4 /D and B 0Wp N 5 /D Where 101 (5-7)

8 10 Journal of Reliability and Statistical Studies June 011 Vol. 4 (1) N p 14 p 4 + p p 16 6 U 1 p U 4 p 4 U 3 p 7 U 4 N 4 p 03 3 U 0 p p U 1 p p U p U 3 p U 4 N 5 p U 1 + p p 7 p p 9 9 U p U p 4 14 U 4 and D is the same as in last section. Now the expected busy period of the repair facility in the repair of compressor blower room and water pump during (0 t) is given by cb (t) t 0 B 0c (u) du Bb (t) t 0 B B0 (u) du Wp and b (t) t B 0Wp (u) du 0 (8-30) 4.4 Cost Benefit Analysis We are now in a position to obtain the profit function by considering mean up time of the system during (0 t) and expected busy period of the repair facility during (0 t). Let us suppose K0 revenue per unit up time K1 payment to repair facility per unit time when repair facility is busy in K the repair of compressor. payment to repair facility per unit time when repair facility is busy in the repair of blower room. payment to repair facility per unit time when repair facility is busy in K3 the repair of water pump. The expected profit incurred by the system during (0 t) is given by P(t) Expected total revenue in (0 t) Expected total repair cost in (0 t) K 0 up (t) K 1 bc (t) K Bb (t) K 3 bwp (t) (31) The expected profit per unit time in a steady state is given by c B wp P K0 A0 K1 B0 K B0 K 3 B0 Where A 0 B 0c B B0 and B 0wp has been already defined. 10 (3)

9 Stochastic Analysis of an Air Condition Particular Case In this section if we consider the case when all repair time distribution are also exponential i.e. G c (t) 1 e c t G b (t) 1 e Then the steady state transition probabilities will be p 10 c p b c w c p 15 b t G w (t) 1 e w t b p 411 b c w c Similarly we can find others. 6. Graphical Representation For more concrete study of the behavior of the characteristics obtained under study in above particular case we plot the curves for reliability and MTSF for different values of b and c while the other parameters are kept fixed as w K b 0.04 c 0.05 K K 150 w 0.08 K 3 50 In Fig.- curves represents the graph of reliability with respect to time for different values of b taken as b and.06. when c.00. It is observed that the reliability decreases as b increases. In Fig. 3 and Fig. 4 curves represent the graphs of MTSF and profit function with respect to b for different values of c and It is observed that the MTSF decreases as b increases. Also with an increase in c the MTSF decreases. Further we observe that the rate of decrement in MTSF is rapid initially and tends to vanish as b becomes large. From Figure-4 it is obvious that the profit decreases with the increase in with b.further with an increase in c there is a decrease in profit with a constant rate. Acknowledgment We sincerely thanks to the Department of Science and Technology (DST) Government of India for financial support under Woman Scientist Scheme-A (SR/WOS-A/MS-09/009) to Punam Phartyal. References Gupta R. and Kumar K. (007). Cost benefit analysis of distillery plant system Int. J. Agricult. Stat. Sciences Vol. 3 No. p Gupta R. and Shivakar (003). Analysis of a stochastic model of cloth weaving system IAPQR Vol. 8(1) p Gupta R. and Chaudhary A. (1994). Profit analysis of a two-unit man-machine system with random appearance and disappearance of the operator Microelectron Reliab. 34(6) p Gupta R. Goel L.R. and Chaudhary A. (1994). Analysis of a two unit standby system with fix allowed down time and truncated exponential life time distribution Reliability Engineering and System Safety Vol. 44 p

10 104 Journal of Reliability and Statistical Studies June 011 Vol. 4 (1) Gupta R. and Goel L.R. (1990). Cost benefit analysis of two-unit parallel standby system administrative delay in repair Int. Jr. of System Science 1 p Arora N. and Kumar D. (000). System analysis and maintenance management for coal handling system in paper plant IJOMAS 16() p Kumar S. Kumar D. and Metha N.P. (1996). Behavioural analysis of a shell gasification and carbon recovery process in a urea fertilizer plant Microelectron. Reliab. 36(5) p TRANSITION DIAGRAM Fig.1 104

11 Stochastic Analysis of an Air Condition 105 R E L IA B IL IT Y R E L A T IO N B E T W E E N R E L IA B IL IT Y A N D T IM E λ ь λ ь λ ь T 80 Fig. Fig

12 106 Journal of Reliability and Statistical Studies June 011 Vol. 4 (1) Fig