A stream W is to be fed as reactant into a reactor $($R$1).$ However, the hot stream $W$ at $200C,$ which is flowed out from a storage tank $($ST$1),$ needs to be cooled to $40C$ before it is fed to the reactor so that overheating in the reactor can be prevented. FIGURE Q$4$ shows the designed cooling system to prevent overheating in the reactor. The cooling system is operated in the manner where the flow control valve $($FCV$1)$ is used to manipulate the flowrate of cooling water through the cooler. TABLE Q$4$ shows the failure probabilities for different components in the system. Develop a fault tree for the top event of "overheating in the reactor $($R$1)"$ by referring to failure probabilities specified by TABLE Q$4$ in the fault tree. Provide all the minimum cut sets from the developed fault tree. Estimate the probability value of the top event for the fault tree. State and justify appropriate assumption.

FIGURE Q$4$: A cooling system for hot stream C before entering reactor $($FCV is flow control valve, FIC is flow indicating controller, FIT is flow indicating transmitter$).$

TABLE Q$4$: Failure Probability Data

$\begin{array}{cc}\text{C}\text{o}\text{m}\text{p}\text{o}\text{n}\text{e}\text{n}\text{t}& \text{F}\text{a}\text{i}\text{l}\text{u}\text{r}\text{e}\text{P}\text{r}\text{o}\text{b}\text{a}\text{b}\text{i}\text{l}\text{i}\text{t}\text{y}\\ \text{A}\text{l}\text{a}\text{r}\text{m}\text{f}\text{a}\text{i}\text{l}\text{s}& 0\mathrm{.}45\\ \text{O}\text{p}\text{e}\text{r}\text{a}\text{t}\text{o}\text{r}\text{f}\text{a}\text{i}\text{l}\text{s}& 0\mathrm{.}08\\ \text{P}\text{u}\text{m}\text{p}(P1)\text{s}\text{t}\text{o}\text{p}\text{s}& 0\mathrm{.}11\\ FCV1\text{f}\text{a}\text{i}\text{l}\text{s}& 0\mathrm{.}30\\ \text{C}\text{o}\text{o}\text{l}\text{i}\text{n}\text{g}\text{t}\text{o}\text{w}\text{e}\text{r}(CT1)\text{f}\text{a}\text{i}\text{l}\text{s}& 0\mathrm{.}22\end{array}$