$04$Figure $1$ shows a simplified flow diagram for a typical chemical process plantQ$4$ Figure $1$ shows a simplified flow diagram for a typical chemical process plant.

In this typical plant both air and reagent gas are passed continuously into a reactor vessel in which a reaction is carried out at a temperature of approximately $1000C$ and atmospheric pressure. The reactor outlet gases are taken through a cooler which allows the unwanted gases to be condensed and drained off, the product gas then being pressurized to about $800k\frac{N}{m^2}$ before passing to stock tanks for subsequent usage in other plants.

Explosive conditions could arise in the plant if:

$(1)$ the temperature in the reactor fell, or

$(2)$ the ratio of the two gases was not maintained within required limits$.$ To safeguard against these hazards an automatic shutdown system is normally installed. This would cater for a divergence from safe to hazardous conditions which may take place within a short time. Consequently little reliance can be placed upon the operator even if the plant were constantly attended. The basic el ements of the shut $-$ down system are shown in Figure $1$ the tempe rature in the reactor is monitored by two measurement lines arranged in a $1-$out$-$of$-2$ logic and a shut down signal is given on a fall in temperature. The composion of the product gas is also monitored on a $2-$out$-$of$-3$ basis. When unacceptable deviations occur in either of these process variables then the plant is automatically shut down by means of signals from the safety circuits operating the shut down valves on a $1-$out$-$of$-2$ basis Assuming that the safety circuits are perfect and that the reliability of the other measurement components are given by R$,$ R$2,$ R$7$ respect ively a$)$ Sketch the reliability block diagram for this system b$)$ What is the probability of an explosion in this chemical plant? $($In terms of R$1,$R$2,...$R$7).$