Chapter-3 PERFORMANCE MEASURES OF A MULTI-EVAPORATOR TYPE COMPRESSOR WITH STANDBY EXPANSION VALVE

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1 Chapter-3 PERFRMANCE MEASURES F A MUTI-EVAPRATR TYPE CMPRESSR WITH STANDBY EXPANSIN VAVE 3. INTRDUCTIN In this model, the author has onsidered a refrigeration plant whih ontains a single ompressor with multi evaporators, for analysis of some important performane measures. These single ompressor type refrigeration plants an be ategories in to following: (i) (ii) (iii) Multi evaporator type at same temperature Two evaporator type at dual temperature Multi evaporator type at multi-temperature The work of evaporator together with expansion valve is to give the onstant temperature orresponding to required state. Thus the different evaporators an be fixed either for same temperature or different temperatures. In this model, the author s investigations are based on multi-evaporator type at same temperature. The whole refrigeration plant is divided in to four subsystems, namely A, B, C and D, onneted in series. The sub system A is ompressor and the subsystem B has three evaporators onneted in parallel. The subsystem C has two expansion valves in standby redundany and the standby expansion valve followed on line through a perfet swithing devie. The subsystem D is ondenser. The system onfiguration has shown in fig.-. The whole system an fail due to failure of its either subsystems. Sine, the system is of non-markovian nature, the author has used supplementary variable tehnique to formulate the mathematial model. aplae transform has been used to solve

2 the symboli model. All the failures follow exponential time distribution where as all the repairs follow general time distribution. This study is divided in to two setions as follows: Setion I Enhaned reliability of refrigeration mahine through standby expansion valve under Head-of-line repair Setion II Profit analysis of refrigeration mahine under Pre-emptive repeat repair Steady state behavior, partiular ase and a numerial example with graphial illustration have appended at the end of eah setion to highlight the important results of the study.

3 System Configuration Evaporator -I Evaporator -II Perfet Swithing Devie Evaporator -III X Expansion Valves X Condenser Compressor Fig-

4 Setion- ENHANCED REIABIITY F REFRIGERATIN MACHINE THRUGH STANDBY EXPANSIN VAVE UNDER HEAD-F-INE REPAIR In this setion, the author has evaluated the reliability and M.T.T.F. of the onsidered refrigeration mahine under head-of-line repair poliy. This poliy is nothing but the first ome first served poliy. The state-transition diagram has shown in fig ASSUMPTINS The following assumptions have been assoiated with this model: (i) (ii) Initially, all the omponents of onsidered system are good. There is one standby expansion valve whih followed online through a perfet swithing devie. (iii) (iv) (v) There is no time lap between failure and start of repair. Head-of-line poliy has been adopted for repair. All the failures follow exponential time distribution whereas all repairs follow general time distribution. (vi) (vii) Failures are statistially independent. Nothing an fail from the failed state. (viii) After repair, system works like a new NTATINS The following notations have been used throughout this model: P (t) : The probability that at time t, the system is in good state of full effiieny. P i (j, t) : The probability that at time t, the system is in failed

5 state due to failure of i th subsystem and elapsed repair time lies in the interval ( j, j + ). Where i A, B, D and j x, y, r respetively. P i (j,t) : The probability that at time t, the system is in failed state due to failure of i th subsystem while one expansion valve has already failed. The elapsed repair time lies in the interval ( j, j + ). P (n,t) / P C (z,t) : The probability that at time t, the system is in Good / failed state due to failure of one / two expansion valves and elapsed repair time lies in the interval (n, n + ) / (z, z + ). a/b//d :Failure rate of subsystem A/B/C/D. α (x) /α 2 (y) / α 3 (n) / α 4 (r) : The first order probability that the subsystem A/B/C /D will be repaired in the time interval (x, x + ) /(y, y + )/ (n, n + ) / (r, r + ), onditioned that it was not repaired up to the time x / y / n / r. α 5 (z) : The first order probability that both the expansion valves will be repaired in the time interval (z, z + ), onditioned that these were not repaired up to the time z. s : aplae transform variable. P (s) : aplae transform of P(t) S i (k) : α k k dk i( ) e α ) i ( for all i and k. D i (k) : - S i ( k) / k for all i and k.

6 R(t) M.T.T.F. : Reliability of the system at time t. : Mean time to failure FRMUATIN F MATHEMATICA MDE By using elementary probability and ontinuity arguments, one an obtain the following set of differene-differential equations governing the nature of onsidered system: d dt z zz + a + b+ + d P () t P A( xt,) α( xdx ) + P B( yt,) α ydy 2 ( ) D B z z + P D ( rt, ) α 4( rdr ) + P ( nt, ) α 3( ndn ) + PC ( zt, ) α 5 ( zdz ) ( ) + + α( x) P A ( xt, ) -----(2) x t + + α 4 ( r) P ( rt, ) (3) r t + + α 2 ( y) P ( yt, ) (4) y t + + a+ b+ + d + α 3( n) PC ( nt, ) ---(5) n t + + α n t C P B C D C ( n) P C A ( n, t) a P ( n, t) ---(6) α 3 ( n) P ( n, t) b n t ( n, t) ---(7) + + α 3 ( n) P ( n, t) d P ( n, t) (8) n t C

7 + + α 5 ( n) P C ( zt, ) (9) z t State-transition Diagram P C B(n,t) b P C (z,t) P C A(n,t) a P C (n,t) d P C D(n,t) α 3 (n) α 3 (n) α 3 (n) P A (x,t) a α (x) P o (t) d α 4 (r) P D (r,t) α 5 (z) b α 2 (y) α 3 (n) P B (y,t) Fig-2

8 Boundary onditions are: z z z P A t a P t PC A n t α 3 n dn (, ) ( ) + (, ) ( ) ( ) P D t d P t PC D n t α 3 n dn (, ) ( ) + (, ) ( ) ( ) P B t b P t PC B n t α 3 n dn (, ) ( ) + (, ) ( ) ( 2) P C A (,t) (3) P C D (,t) (4) P C B (,t) (5) P C (,t) P C (t) (6) P C (,t) P (t) (7) Initial onditions are: P ( ), otherwise zero (8) 3..4 SUTIN F MDE Taking aplae transforms of equations () through (7) by making use of initial onditions (8), we have: z zz s+ a+ b+ + d P() s + PA(,) xsα() xdx + PB( ys,) α2 ( ydy ) z z + PD( r, s) α4( r) dr + P ( ns, ) α3( ndn ) + PC( z, s) α 5( z) dz ( 9) + s+ α( x) P A ( xs, ) -----(2) x

9 B + s+ α 4 ( r) P D ( r, s) (2) r + s+ α 2 ( y) P ( y, s) (22) y + s+ a+ b+ + d + α 3( n) P ( ns, ) ---(23) n + s+ α 3 ( n) P A ( n, s) a P ( n, s) (24) n n B D + s+ α 3 ( n) P ( n, s) b P ( n, s) ---(25) + s+ α 3 ( n) P ( n, s) d P ( n, s) (26) n + s+ α 5 ( z) P C ( z, s) ---(27) z z PA(, s) a P( s) + P A( ns, ) α 3 ( ndn ) ( 28) z PD(, s) d P( s) + P D( ns, ) α 3 ( ndn ) ( 29) z PB(, s) b P( s) + P B( ns, ) α 3 ( ndn ) ( 3) P A (, s ) ( 3 ) P D (, s ) ( 3 2 ) P B (, s ) ( 3 3)

10 P C (, s) P ( s) and P (, s) P ( s) ---(34) (35) Now integrating (23) by using (35), we get -(s+a+b++d)n- 3 dn P ( ns, ) P ( s) e zα (n) P () s P () s D 3 (s + a + b + + d) (36) Integrating (24) by making use of (3), we have apo( s) P n s sn n dn s a b d n n dn A (, ) e α 3( ) ( ) e α 3( ) a + b + + d P apo() s () s [ a b d D ( s) D ( s+ a+ b+ + d)] A 3 3 z z ap() s E ( say) ( 37) Similarly, equations (25), (26) give on integration, by using (33) and (32), respetively P B ( s) b P ( s) E ( 38) P D ( s) d P ( s) E ---(39) Again, equations (27) gives on integration together with (34): sz P (,) zs P () s () zdz e zα 5 () () ( ) () ( ) 2 P s P s D3 s a b d D5 s 4 Now by using relevant relations, equation (2) gives on integration:

11 sx P xs A P os xdx (, ) (, ) α ( ) A e P ( s ) A P ( os, ) A D( s) z a P ( s) + { a b d S ( s ) S ( s + a + b + + d )} D ( s) ab P ( s) D ( s) ( say) ( 4) Similarly, equations (2) and (22) give on integration, respetively: PD ( s) d P ( s) B D 4(s) ( 42) and PB ( s) b P ( s) B D 2 (s) (43) astly, equation (9) beomes by making use of relevant relations: P () s As () Thus, finally we have the following aplae transforms of different state probabilities: P ( s ) A ( s ) ( 44) P A ( s) ab D ( s ) ( 45) A ( s) P D ( s) db D 4 ( s ) ( 46) A ( s) P B ( s) bb D 2 ( s ) ( 47) A ( s) P ( s) D 3( a + b + + d) ( 48) A( s)

12 P A ( s ) a E A ( s) ( 49) P B ( s ) b E A ( s) ( 5) P D ( s ) d E A ( s) ( 5) P C ( s) 2 A( s) D 3( s+ a + b + + d) D 5( s) ( 52) where, D i ( s) Si ( s) i 2, 5 s B + a + b + + d 3 3 { S ( s) S ( s + a + b + + d )} E a + b + + d [ D 3( s ) D 3( s + a + b + + d )] and A ( s) s + a + b + + d S 3 ( s + a + b + + d ) 2 D 3 ( s + a + b + + d ) S 5 ( s) B[ a S ( s) + d S 4( s) + b S 2( s)] 3..5 STEADY-STATE BEHAVIR F THE SYSTEM lim ( lim t s Using Abel s lemma, viz ; P t) s P ( s) P ( say), provided the limit on R.H.S. exists, we have the following time independent state probabilities from equations (44) through (52): P ( 53) A '( )

13 ab P A M A '( ) db P D M 4 A '( ) bb P B M 2 A '( ) ( 54) ( 55) ( 56) P P P P D ( a+ b+ + d) A'( ) ( 57) a A '( ) E ( 58) b A '( ) E ( 59) d A '( ) E ( 6) 3 A B D 2 PC D3( a+ b+ + d) M5 ( 6) A'( ) where, Mi S '( ) i 2, 5 d A'( ) ds As ( ) i s and B E + [ S ( a + b + + d )] a + b + + d [ M D ( a + b + + d )] a + b + + d PARTICUAR CASE When repairs follow exponential time distribution Setting S i() s α i/ s + α i i 2,, 5 et., we have from equations (44) through (52), the following aplae transforms of different state probabilities in this ase:

14 P ( s) ( 62) F( s) P A ( s) ag F ( s) s + α ( 63) P D () s dg Fs () s+ α 4 ( 64) P B ( s) bg F ( s) s + α 2 ( 65) P ( s) F ( s) s + a + b + + d + α 3 ( 66) P A ( s ) a F ( s) H ( 67) P B ( s ) b F ( s) H ( 68) P D ( s ) d F ( s) H ( 69) PC () s where, 2 F() s ( s+ a+ b+ + d+ α3)( s+ α 5) ( 7) G H α ( s + α 3)( s + a + b + + d + α 3 ) ( s + α 3)( s + a + b + + d + α 3) α H 3 and F ( s) s + a + b + + d G a α + s + α d α 4 s + α 4 s + a + b + + d + α b α 2 + s + α α 5 s + α 5

15 3..7 REIABIITY ANAYSIS We have R ( s) s + a + b + + d on inverting this, w e have R(t) exp.{-(a + b + + d)t} ( 7) Again, M.T.T.F. lim Rs () s ( 72) a + b + + d 3..8 NUMERICA CMPUTATIN For a numerial omputation, let us onsider the values a., b.2,.3, d.4 and t,, By using these values in relations (7) and (72), one an observe the hange in orresponding performane measure, with respet to time. These hanges have shown in table () and (2), also orresponding graphs have shown in figs.(3) and (4) respetively RESUT & DISCUSSIN We plot two graphs shown in the figs (3) and (4) and the orresponding values are given in the tables (), (2) respetively. These figs show the hanges in different performane measures of the onsidered system with respet to hosen parameters. The analysis of fig (3) reveals that reliability of the system dereases rapidly for lower values of time t but after t5 it dereases appx. in uniform manner. A ritial examination of fig (4) reveals that, as we make inrease in failure rate a, M.T.T.F. of the system dereases in a onstant manner.

16 t R(t) x -5 Table- Reliability Vs Time---> Series Time--> Fig-3

17 a M.T.T.F Table-2 MTTF Vs a ---->.5..5 Series a ---> Fig-4