Availability based operational behavior of B-Pan crystallization system in the sugar industry

Abstract

In this paper, the B-Pan crystallization system in the sugar industry is analyzed that having three subsystems—crystallizer, centrifugal machine and melter. These subsystems are arranged in series. Subsystem ‘centrifugal machine’ has standby unit, subsystem 'crystallizer' has four parallel units and subsystem ‘melter’ has single unit. The time to failure of the subsystems follows negative exponential distribution while repair time distribution is taken as arbitrary. The differential–difference equations of the system model are developed by using supplementary variable technique and solved to get state transition probabilities of B-Pan crystallization system. A particular case is also considered to show the behavior of availability and expected profit of the B-Pan crystallization system in the sugar industry.

Appendex 1

Supplementary variable technique

When the repair rate or failure rate or both are time-dependent, the system loses its Markovian character, in this situation; the future event will not depend on present only (like Markov events) but will depend past also. These events are known as non-Markovian events.

As this paper discusses B-Pan crystallization system of the sugar industry consisting system consists of three subsystems namely crystallizer, centrifugal machine and melter with constant failure rates and arbitrary repair rates so that now the system is of non-Markovian nature. By introducing a new variable, called supplementary variable, the non-Markovian nature of the system is changed to Markovian.

Here we introduced variable x (as a supplementary variable) now the nature of the system becomes Markovian, e.g. the equation number 1(having x as a supplementary variable):

$$p_{0} (t + \Delta t) = [1 - \alpha_{1} \Delta t - \alpha_{2} \Delta t - \alpha_{3} \Delta t]p_{0} (t) + \int\limits_{0}^{\infty } {\beta_{1} (x)p_{1} (x,t)dx} + \int\limits_{0}^{\infty } {\beta_{2} (x)p_{2} (x,t)dx + \int\limits_{0}^{\infty } {\beta_{3} (x)p_{4} (x,t)dx} }$$

Dividing both sides by ∆t

$$\frac{{p_{0} (t + \Delta t) - p_{0} (t)}}{\Delta t} = [ - \alpha_{1} - \alpha_{2} - \alpha_{3} ]p_{0} (t) + \int\limits_{0}^{\infty } {\beta_{1} (x)p_{1} (x,t)dx} + \int\limits_{0}^{\infty } {\beta_{2} (x)p_{2} (x,t)dx + \int\limits_{0}^{\infty } {\beta_{3} (x)p_{4} (x,t)dx} }$$

As $\Delta t \to 0$

$$\left[ {\frac{d}{dt} + \alpha_{1} + \alpha_{2} + \alpha_{3} } \right]p_{0} (t) = \int\limits_{0}^{\infty } {\beta_{1} (x)p_{1} (x,t)dx} + \int\limits_{0}^{\infty } {\beta_{2} (x)p_{2} (x,t)dx} + \int\limits_{0}^{\infty } {\beta_{3} (x)p_{4} (x,t)} dx$$

(1)

In this way, with the help of supplementary variable technique we can obtain various differential difference Eqs. (1–18).