Operational behaviour and profit function for a bleaching and screening system in the paper industry

Microelectron. Reliab., Vol. 33, No. 8, pp. 1101-1105, 1993.

0026-2714/9356.00+ .00 © 1993PergamonPress Ltd

Printed in Great Britain.

OPERATIONAL BEHAVIOUR A N D PROFIT FUNCTION FOR A BLEACHING A N D SCREENING SYSTEM IN THE PAPER INDUSTRY DINESH KUMAR Mechanical Engineering Department, Regional Engineering College, Kurukshetra 132 119, India JAI SINGH Mathematics Department, Regional Engineering College, Kurukshetra 132 119, India and P. C. PANDEY Mechanical and Industrial Engineering Department, University of Roorkee, Roorkee 247 667, India

(Received for publication 11 March 1992) Abstract--This paper discusses the bleaching and screening system (each having four subsystems) in the paper industry with three states: good, reduced and failed. The failure rate for each subsystem is constant while the repair rates are arbitrary. Mathematical formulation is carried out using the supplementary variable technique. The equations are solved using direct integration method. Expressions and graphs for steady state availability and mean time to system failure (MTSF) are given, depicting the effect of failure and repair rates. Profit analysis is also carried out.

INTRODUCTION

NOTATION

The goal of m a x i m u m production and long-run availability under the given operative conditions may be achieved, making the system failure free (as far as possible), by proper maintenance planning. A probabilistic analysis of the system under given operative conditions is helpful in designing and maintenance of each equipment. In the paper industry we have various systems, namely chipping, feeding, pulp formation, washing, bleaching, screening, paper formation and collection. The important processes of the paper industry, upon which the quality of paper and hence profit depends, are the bleaching and screening processes. In the process of paper formation, the chips from storage are fed into a digester to form the pulp which is processed through systems called knotter, decker, opener and washing. These systems have been discussed in detail in [1-5]. The washed pulp is kept in a chamber where chlorine, at a controlled rate, is pressed through the pulp for a few hours. The pulp is passed over a filter and opener in two stages to get chlorine-free white pulp. This pulp is processed through filter, screen, cleaner and decker arranged in series for final removal of chlorine, knots and other undesirable materials. The pulp so prepared is stored in a chamber. This paper discusses the behaviour of the two systems--"Bleaching" and "Screening".

Gt, H t, A, B, D, E

capital letters are used for units working in full capacity, bar on capital letters is used for reduced state and small letters for respective failed state ~j respective constant failure rate of subsystems Gt,/-/i, A, B, and E (j = 1, 2, 3, 4, 5) ).t failure rate of subsystem D from fully operating state to reduced state 2 failure rate of subsystem D from reduced state to failed state flj(xj) j = 1, 2. . . . . 6 respective variable repair rate of subsystems Gt, nt, A, B, D and E Po(t),P(t) probability that bleaching, screening system is working in full capacity Pr(x, t) probability density that the system is in r = 1, 2 . . . . ,15 rth state and the elapsed repair time is x Dashes on P represents the first derivative. Limit of integration is from 0 to oo. Assumptions used are same as in Ref. [5]. SYSTEM DESCRIPTION The bleaching system has four subsystems G~ and H,; 1 = 1, 2 of the same nature and capacity. GI, H l and G 2,//2 are in series. (i) The filter (Gt), the opener (HI) each comprises of two parallel units. Failure of even one unit fails the system due to a fall in quality of the processed pulp.

1101

D. KUMARet

1102

(ii) The filter (Al), the screen (A2) and decker (A3) each has single unit in series. Failure of any one unit causes complete system failure. (iii) The cleaner (D) comprises of three units in parallel. Failure of one, two or three units reduces the capacity of the system forcing to carry the process manually. The nature and behaviour of each subsystem may be tabulated as below.

Behaviour

al.

and is given in [5]. Since the bleaching system and screening systems are in series, the reliability of the combined system at any time " t " is given by

R(t) = R,(t). Rn(t ).

The mean time to system failure, MTSF, is obtained through

fo ~ tR(t) dt,

Effect/occurrence

Subsystem Gt, A t and A2: Major failure

Remedy

Rare

Minor failure Performance degradation

Often Yes

Inconvenience in operation

Yes

By regular check of bearings, lubrication and oil level in gear box Repaired in a short time by a skilled worker Can be rectified with regular watch and repaired by skilled worker only Maintain the vacuum (providing vacuum pump in standby).

Subsystem H: Major failure Minor failure Performance degradation Inconvenience in operation

Very rare Rare Possible None

Shut down the unit and repair (repair takes a long time) Easily repairable in a short time by skilled worker Repair is the only remedy

Subsystem D: Major failure

Very rare

Bypass the unit and repair by an unskilled worker in short time Avoid through regular watch

Minor failure Performance degradation Inconvenience in operation

Often None None

Subsystem A3: Major failure Minor failure Performance degradation Inconvenience in operation

Rare Often None Very large

Shut down the unit and repair takes long time Shut down the unit and repair in short time Control the vacuum

Transition diagrams for the system are shown in Figs 1 and 2.

considering that a constant repair rate in each subsystem M T S F is given as: MTSF =

ANALYSIS OF THE SYSTEM

~61L[ 1 +j_~-~5fl~G(J (AJr f16)(f12~Jr fl3)3 -/[Q]_~-1. (4)

(a) Differential equations associated with Fig. 1 are: P~(t) + 2,=1 ~ ~t'P°(t) = ,=1 ~" f

P,(x,, t)fl](x,)dx 1

+~=3fP,(x2, t)fl2(x2)dx2

gl HI I

(1) %

v.+~ +Mx')

l=2,

m =3,4;

with initial and boundary conditions

P,,(0, t) = ~ttP0(t);

otherwise = O; l=1, •=2,

I~1(xl)

(,,)

(2)

P0 (0) = 1

(3)

132(x2)

%

m = 1,2; m=3,4.

(b) Differential equations associated with Fig. 2 are given in [5]. The reliability R,(t) of the screening system is

R,(t) = P(t) + Ps(x6, t) + P9(x6, t)

0

Systemis good

[ Fig. 1.

[ System in failed state

Operational behaviour and profit function

aiD

1103

in each differential equation and on solving, the combined system availability A v is obtained as

I

5 0C.

1

(5) where 2c(2 Q

=

1

~3]

.

BEHAVIOUR ANALYSIS

la,°l O

la, l

System is good

System in reduced state ,~ [

System in more reducedstate I System in failed state Fig. 2.

Since the filters in the bleaching and in screening systems are of the same nature, we may take ~l = ct3; fll = f13. F o r the steady state availability of the plant, i.e. t~oo,

d ~0,

d dt=O,

~=0

we put

Taking values of fl2=0.2, fl3=0.2, fl4=0.2, f15 = 0.2 and f16 = 0.5, the effect of the failure rate is tabulated in Tables 1-6. F r o m Tables 1 and 2, we can judge the nature of the opener, decker and filter and its effect on availability and MTSF. It shows that even if the failure rates of opener, decker and filter are zero (which is practically not possible) the availability of the system is about 95%. Taking all these factors into consideration the lower limit is assumed to be 70%. If the availability of the system goes below this, then there may be a loss in profit, and hence the system requires thorough checking of the equipment. Table 3 clearly shows the effect of the failure rate of the filter and screen. The minimum failure rate of filter is once in two hundred hours which forces the failure rate of the screen to be maintained, i.e. it must be within the limit of once in two hundred hours and also has to reduce the failure rate of decker from three times in one hundred hours to twice in one hundred hours by proper maintenance planning. Taking the failure rate as discussed above, the availability of the system is maintained at about 70%.

Table I. Effect of failure rate of opener, decker and filter (taking 54 = 0.1, 2 = 0.005) Availability (A,,) 52 0.0 0.025 0.05

55

53=0.0

0.0 0.025 0.0 0.025 0.0 0.025

0.9524 0.8511 0.7619 0.6809 0.6349 0.5674

cq=0.02 53=0.04 53=0.06 53=0.08 53=0.10 0.7246 0.6536 0.5997 0.5409 0.5115 0.4614

0.5714 0.5195 0.4848 0.4407 0.4215 0.3828

0.4630 0.4237 0.4004 0.3665 0.3527 0.3228

0.3831 0.3527 0.3364 0.3097 0.2999 0.2761

0.3226 0.2985 0.2867 0.2653 0.2581 0.2388

Table 2. Effect of failure rate of opener, decker and filter (taking 54 = 0.1, 2 = 0.005) MTSF 52

55

53 = 0.0

53 = 0.025

53 = 0.05

53 = 0.075

53 = 0.1

0.0

0.0 0.05 0.1 0.0 0.05 0.1 0.0 0.05 0.1

1430 1440 1630 953.3 960 1086.7 715 720 815

1184 1192 1344 845.7 851.4 960 657.7 657.7 746.6

1020 1026.6 1153 765 770 865 612 616 692

902.8 908.6 1017.1 702.2 706 791.1 574.5 578.2 647.2

815 820 915 652 656 732 543.3 546 610

0.05 0.1

D. KUMAR et

1104

al.

Table 3. Effect of failure rate of screen and filter (taking ~2 = 0.01, c%= 0.03) Availability (A~) u4 0.0 0.01 0.03 0.05

a 3 = 0.0 a3 = 0.02 a 3 = 0.04 ~3 = 0.06 a 3 = 0.08 ~3 = 0.1 0.7246 0.6944 0.6410 0.5952

0.5714 0.5495 0.5102 0.4762

0.4630 0.4464 0.4167 0.3906

0.3831 0.3704 0.3472 0.3268

0.3256 0.3125 0.2941 0.2778

0.2755 0.2674 0.2525 0.2392

Table 4. Effect of failure and repair rate of screen and filter (taking ~2 = 0.01, ~5 = 0,03)

MTSF f14

~4

~3=0.0

0.01

0.01 0.025 0.05 0.01 0.025 0.05

935 1235 1735 775 835 935

0.05

~3=0-25 ~3=0.05 853.3 1120 1564.4 711.1 764.4 853.3

Tables 5 a n d 6 show t h a t the filter repair rate has the least effect o n availability while the increase in the repair rate of the decker from 0.01 to 0.1 increases availability significantly. Also, the increase in the repair rate o f the opener from 0.1 to 0.25 increases the availability sharply a n d the repair rate of the screen from 0.01 to 0.3 increases availability by a considerable degree. The a b o v e tables clearly indicate the level of the repair rate a n d the possibility o f failure o f equipment. This helps in m a i n t a i n i n g the repair rate o f each piece o f e q u i p m e n t so t h a t the availability o f the system can be m a i n t a i n e d at a level giving profit to the industry.

788 1028 1428 660 708 788

~3=0.075

%=0.1

734.5 952.7 1316.4 618.2 661.8 734.5

690 890 1223.3 583.3 623.3 690

E2 = fll f12 + 2(cq f12 + ~2 fit);

Go= Gl

=

0102 ' 02 -- (ill+ f12)0~+ fltfl2

01(01 G2

02)

02 (02 - 01 )

If the failure rate of the cleaner m a y be m a d e negligible, providing a n unskilled worker o n cleaner

(D), i.e., 2 -~ 0, 21 ~ 0. W i t h a c o n s t a n t repair rate the reliability function for the screening system is given in [5], as

Rs(t) PARTICULAR CASE

R~(t) =fllfl2fl3 + Kle-bl + K2e-a-b K3e -d'. bcd

W i t h a c o n s t a n t repair rate, the value of the reliability function RB(t ) for the bleaching system is given by

RB(t)=Go+Gle-°~' +G2e-°2%

(6)

where

-El ___E,,//~2-4~2. 01'2 =

2

PROFIT ANALYSIS Considering a n unskilled worker o n cleaner (D), the reliability of the bleaching a n d screening system is given by

'

R(t) = (Go + GI e -°''+ El-~-~l'-l-~2"[-fll'-l-fl2

Availability (Av)

0.25 0.5

f15 0.01 0.1 0.2 0.01 0.1 0.2 0.01 0.1 0.2

f13 =0.I 0.2151 0.5128 0.5848 0.3779 0.6105 0.6962 0.4036 0.6519 0.7434

f13 =0.2 0.2243 0.5409 0.6185 0.3983 0.6482 0.7412 0.4265 0.6941 0.7937

G2

\ bcd

;

+ Kt e-bt + K2 e-C' + K3 e-d').

Table 5. Effect of failure rate of opener, decker and filter (taking ~t3 = ~t4 = 0.005, ct2 = ~q = 0.02) f12 0.1

(7)

f13=0.3 0.2275 0.5508 0.6304 0.4055 0.6616 0.7572 0.4346 0.7091 0.8116

f13 =0.4 0.2291 0.5559 0.6365 0.4092 0.6684 0.7654 0.4388 0.7168 0.8208

f13 =0.5 0.2301 0.5589 0.6402 0.4114 0.6726 0.7704 0.4413 0.7215 0.8264

(8)

Table 6. Effect of repair rate of screen and filter (taking f12= 0.25, f15= 0.2) Availability (Av)

f14

f13 =0.1

f13 =0.2

f13 =0.3

f13 =0-4

f13 =0.5

0.01 0.03 0.05 0.07 0.09 0.10

0.4810 0.6028 0.6349 0.6498 0.6568 0.6614

0.5086 0.6398 0.6747 0.6908 0.7000 0.7034

0.5184 0.6530 0.6888 0.7054 0.7149 0.7183

0.5233 0.6597 0.6960 0.7128 0.7224 0.7259

0.5264 0.6638 0.7004 0.7173 0.7271 0.7306

Operational behaviour and profit function If the service facility is always available, it remains busy for a time " t " during the interval (0, t]. Let D~ a n d / ) 2 be the revenue per unit time and service cost per unit time, respectively. Also let D 3 and D4 be the standby cost and unskilled worker cost, respectively, then the profit function H ( t ) for interval (0, t] is given as follows: H(t)

D~ .f/ R(t) dt - D 2t - (D 3 --I-D4)

(9)

putting d H ( t ) / d t = 0, we get t =Iog~(GoGIG2KoKIK2K3D2) O/<°'+°2+b+c+d)) (10) Taking the second derivative of equation (9) and putting in the values of " t " from (10), we find that equation (10) gives time for m i n i m u m profit. Hence to get more profit, the plant should be run for a time greater than " t " given by (10). The greater the running time, the greater the profit.

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Acknowledgement--The first author is grateful to the University Grants Commission, New Delhi (India) for financial assistance for this work. REFERENCES

1. Dinesh Kumar, I. P. Singh and Jai Singh, Reliability analysis of the feeding system in the paper industry, Mieroelectron. Reliab. 28, 213-215 (1988). 2. Dinesh Kumar, Jai Singh and P. C. Pandey, Availability analysis of feeding system in the paper industry with general repair time, Proe. Int. Conf. Math. Modelling Sci. Tech. 2. World Scientific, Singapore (1988). 3. Dinesh Kumar, Jai Singh and P. C. Pandey, Maintenance planning for pulping system in paper industry, Reliab. Engng Syst. Safety 25, 293-302 (1989). 4. Dinesh Kumar, Jai Singh and P. C. Pandey, Availablity of washing system in paper industry, Microelectron. Reliab. 29, 775-778 (1989). 5. Dinesh Kumar, Jai Singh and P. C. Pandey, Cost analysis of a multi-component screening system in paper industry, Microelectron. Reliab. 30, 457-461 (t990).