Availability analysis of cattle feed plant using matlab-tool.

Garg, Deepika ; Kumar, Kuldeep

Introduction

Recently, many researchers have discussed reliability of different process industries using different techniques. Kumar and Singh [2] analyzed the Availability of a washing system of paper industry. Singh, Kumar and Pandey [3, 5] discussed the reliability and availability of Fertilizer and Sugar industry. Dayal and singh[4] studied reliability analysis of a system in a fluctuating enviroment. Zaho [6]developed a generalized availability model for repairable component and series system including perfect and imperfect repair. Michelson [7] discussed the use of reliability technology in process industry. Singh and Mahajan [8] examined the reliability and long run availability of a Utensils Manufacturing Plant using Laplace transforms. Gunes and Deveci [9] have studied the reliability of service systems and its application in student office and Habchi [10] discussed and improved the method of reliability assessment for suspended test and Jain [11] discussed N-Policy for redundant repairable system with additional repairman. Gupta, Lal, Sharma and Singh [12] discussed the reliability, long term availability and MTBF of cement industry with the help of Runga-Kutta method. Kiureghian and Ditlevson [13] analyzed the availability, reliability & downtime of system with repairable components. Kumar,Sigh,Sharma [15] discussed the of availability an automobile system namely "scooty". Tewari, Kumar, Kajal, Khanduja [16] discussed the availability of a Crystallization unit of a sugar plant. This paper analyze how availability of cattle feed plant varies with time.

The System

Cattle feed plant mainly consists of seven subsystems namely Elevator, Grinder, Hopper, Mixer, Winch, Palletiser, Screw conveyor. Initially Elevator lifts the material and put it into the Grinder. Grinder grinds the raw material and then the material is put into the Hopper. Hopper is used for the storage and cooling of material by the fans present in the Hopper. Then the material is put into the Mixer for proper mixing of certain additives in specified ratio. This mixture is lift by Winch. Which put this mixture into the Palletiser. Palletiser allows the mixture to move forward and passes through the holes and produces the mandatory shape to mixture. Finally Screw conveyor carries the final product to the store where it is packed for final delivery.

The Cattle feed plant consists of the following seven main subsystems:

(i) Elevator (A) consists of one unit. The system fails when this subsystem fails.

(ii) Grinder (B) consists of one unit. It is subjected to major failure only.

(iii) Hopper (C) consists of one unit. It is subjected to major failure only.

(iv) Mixer (D) consists of two units, one working and the other is in cold standby. The cold standby unit is of lower capacity. The system works on standby unit in reduced capacity. Complete failure occurs when both units fail.

(v) Winch (E) consists of one unit. The system fails when this subsystem fails

(vi) Palletiser (F) consists of two units, one working and the other is in cold standby. The cold standby unit is of lower capacity. The system works on standby unit in reduced capacity. Complete failure occurs when both units fail.

(vii) Screw conveyor (G) consists of one unit. The system fails when this subsystem fails.

Assumptions and Notations

(i) Repair rates and failure rates are negative exponential and independent of each other.

(ii) Not more than one failure occurs at a time.

(iii) A repaired unit is, performance wise, as good as new.

(iv) The subsystems D and F fail through reduced states.

(v) Switch-over devices are perfect.

A, B, C, D, E, F, G : Capital letters are used for good states.

D, F : Denotes the reduced capacity states.

a, b, c, d, e, f, g : Denotes the respective failed states.

[[lambda].sub.i] : Indicates the respective mean failure rates of Elevator, Grinder, Hopper, Mixer, Winch, Palletiser,Screw conveyor. i =1,2,3, 4,5,6,7,8,9. i = 5 and 8 stands for failure rates of reduced states of D and F respectively.

[[mu].sub.i] : Indicates the respective repair rates of Elevator, Grinder, Hopper,Mixer, Winch, Palletiser, Screw conveyor, i =1,2,3,4,5,6, 7,8,9. i = 5 and 8 stands for repair rates of reduced states of D and F respectively.

[P.sub.i] (t) : Probability that the system is in ith state at time t.

[P.sub.i]' (t) : Derivative of probability function Pi(t).

Mathematical Modeling

Probabilistic considerations gives the following differential equations, associated with the transition diagram as given by figure 2.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Availability Analysis

Equations are solved using the Matlab-program, for details of program see the appendix and results are shown below.

Conclusion

Table 1 and figure 1 shows the variation of Availability with respect to time. Initially Availability decreases sharply and then become almost stable after long duration of time.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Appendix

Matlab program
 options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-5]);
 [T,Y] = ode45(@Cattleplant,[0 20 40 60 80 100 120 140 160],
 [1 zeros(1,27)]);
 Sumy=sum(Y(:,1:4)')';
 plot(T,Sumy);
Subprogramme
 function dy =Cattleplant(t,y)
 dy = zeros(28,1); % a column vector
 a1=0.001;
 a2=0.001;
 a3=0.004; a4=0.0025; a5=0.0025; a6=0.006; a7=0.005; a8=0.005;
 a9=0.003;
b1=0.01; b2=0.04; b3=0.02; b4=0.02; b5=0.02; b6=0.06;b7=0.05;b8=0.05;
b9=0.03;
 dy(1) = -
(a1+a2+a3+a4+a6+a7+a9)*y(1)+b1*y(5)+b2*y(6)+b3*y(7)+b6*y(8)+b9*y(9)+
b4*y(4)+b7*y(2);
 dy(2) = -
(a1+a2+a3+a4+a6+a8+a9+b7)*y(2)+b1*y(21)+b2*y(20)+b3*y(19)+b4*y(3)+b6*y
(18)+b8*y(17)+b9*y(16)++a7*y(1);
 dy(3)= -
(a1+a2+a3+a5+a6+a8+a9+b4+b7)*y(3)+b1*y(28)+b2*y(27)+b3*y(26)+b5*y(25)+
b6*y(24)+b8*y(23)+b9*y(22)+a4*y(2)+a7*y(4);
 dy(4)= -
(a1+a2+a3+a5+a6+a9+a7+b4)*y(4)+b1*y(15)+b2*y(14)+b3*y(13)+b5*y(12)+b6*
y(11)+b9*y(10)+b7*y(3)+a4*y(1);
 dy(5) = -b1*y(5)+a1*y(1);
 dy(6) = -b2*y(6)+a2*y(1);
 dy(7) = -b3*y(7)+a3*y(1);
 dy(8) = -b6*y(8)+a6*y(1);
 dy(9) = -b9*y(9)+a9*y(1);
 dy(10) = -b9*y(10)+a9*y(4);
 dy(11) = -b6*y(11)+a6*y(4);
 dy(12) = -b5*y(12)+a5*y(4);
 dy(13) = -b3*y(13)+a3*y(4);
 dy(14) = -b2*y(14)+a2*y(4);
 dy(15) = -b1*y(15)+a1*y(4);
 dy(16) = -b9*y(16)+a9*y(2);
 dy(17) = -b8*y(17)+a8*y(2);
 dy(18) = -b6*y(18)+a6*y(2);
 dy(19) = -b3*y(19)+a3*y(2);
 dy(20) = -b2*y(20)+a2*y(2);
 dy(21) = -b1*y(21)+a1*y(2);
 dy(22) = -b9*y(22)+a9*y(3);
 dy(23) = -b8*y(23)+a8*y(3);
 dy(24) = -b6*y(24)+a6*y(3);
 dy(25) = -b5*y(25)+a5*y(3);
 dy(26) = -b3*y(26)+a3*y(3);
 dy(27) = -b2*y(27)+a2*y(3);
 dy(28) = -b1*y(28)+a1*y(3);

References

[1] Dhillon, B.S., and Singh, C., 1981, "Engineering Reliability New Techniques and Applications", John Wiley and sons.

[2] Kumar, D.and Singh, J., 1989,."Availability of a Washing System in the Paper Industry", Microelectron Reliability, 29, pp.775-778.

[3] Singh, J., Pandey, P.C. and Kumar, D.,1990, "Designing for Reliable Operation of Urea Synthesis in the Fertilizer Industry", Microelectron. Reliability, 30, pp.1021-1024.

[4] Dayal, B.and Singh, J., 1992, "Reliability analysis of a system in a fluctuating envioronment", Microelectron Reliability, 32, pp.601-603.

[5] Kumar, D., Singh, J.and. Pandey, P.C., 1992, "Availability of the Crystallization System in the Sugar Industry under Common-Cause Failure", IEEE Transactions on Reliability, 41(1), pp 85-91.

[6] Zhao, M., 1994,"Availability for Repairable components and series systems", IEEE Transactions On Reliability, 43(2).

[7] Michelson Q., 1998, "Use of Reliability Technology in The Process Industry", Reliability Engineering and system safety, 60,pp.179-181.

[8] Singh, J. and Mahajan, P., 1999, "Reliability of Utensils Manufacturing Plant--A Case Study", Opsearch, 36(.3), pp 260-269.

[9] Guines, M. and Deveci, I., 2002, "Reliability of service system and an application in student office", International Journal of Quality & Reliability Management, 19,pp.206-211.

[10] G. Habchi, 2002, "An improved method of reliability assessment for suspended tests", International Journal of Quality & Reliability Management, 19, pp.454-470.

[11] Jain Madhu, 2003, "N-Policy for redundant repairable system with additional repairmen", Opsearch, 4, 97- 114

[12] Gupta, P., Lal,A.K., Sharma, R.K. and Singh, J., 2005, "Behavioral Study of the Cement manufacturing Plant-A Numerical Approach", Journal of Mathematics and Systems Sciences, 1(1), pp.50-69.

[13] Kiureghian, A.D. and Ditlevson, O.D., 2007, "Availability, Reliability & downtime of system with repairable components", Reliability Engineering and System Safety, 92(2), pp. 66-72.

[14] Gurjar, Jai Singh, 2007, "Reliability Technology-Theory And Applications"; I.K. International Publishing House Pvt. Ltd. New Delhi (I ndia).

[15] Singh, J., Kumar, K., Sharma, A., 2008, "Availability Evaluation of an Automobile System", Journal of mathematics and system sciences, 4(2), pp.95-102.

[16] Tewari, P.C., Kumar. D., Kajal, S. Khanduja, R., 2008, "Decision support system for the Crystallization unit of a sugar plant", Icfai J. of Science and Technology, 4(3), pp.7- 16.

Deepika Garg (1) and Kuldeep Kumar (2)

(1) Reaseach scholar associated with Dr. Kuldeep Kumar, Dept of Mathematics, N.I.T., Kurukshetra,,Haryana, India. E-Mail ID: deepikanit@yahoo.in

(2) Prof. and Chairman, Dept of Mathematics, N.I.T. Kurukshetra, India E-Mail ID: kuldeepnitk@yahoo.com

Table 1: Variation of Availability With Respect To Time.

Time   Availability

  0        1
 20      0.81001
 40      0.73369
 60      0.6983
 80      0.67965
100      0.66882
120      0.66208
140      0.65767
160      0.65467