You serve as a chemical engineer at the BP refinery, with responsibility for overseeing operations involving the fractionating column. The column receives an equimolar mixture of Benzene and Toluene and consists of $5$ plates, including a condenser and reboiler, with a total condenser. This system maintains a constant volume and integrates a heating coil to warm the crude oil, alongside a shaft for agitating the oil within the tank. Designed to yield a distillate of $95\%$ purity with a $10\%$ waste in the bottoms, the column operates with a relative volatility of $3\mathrm{.}2$ and a reflux ratio of $3\mathrm{.}5.$ The feed remains in liquid state at its boiling point, and reflux is reintroduced to the column at $100$ degrees Celsius. The maximum feed rate is set at $10,000$ kg$/$h$,$ with the minimum feed rate established at $70\%$ of this maximum. Sieve plates are employed in the column.

Upon receiving notification from the refinery manager that the gas plant is non$-$operational, you are tasked with troubleshooting the fractionating column. Subsequent inquiries with unit operating personnel reveal that the pumparound circulating pump is malfunctioning. Whenever the hot oil flow to the reboiler is increased, destabilization occurs within the gas plant. Reboiler heat$-$duty and reflux rates become erratic, and the discharge pressure of the hot$-$oil circulating pump exhibits significant fluctuations. The operating personnel suggest that a replacement pump with reduced net positive suction head may be necessary.

As an engineer, your approach to solving this problem and restoring plant operation involves thorough assessment of the pump malfunction and subsequent actions to address the issue. This includes evaluating the pump's and column$$s performance, considering replacement options, optimizing operating parameters, and collaborating closely with the operations team to ensure effective resolution and sustained stability in plant operations. Show all steps and determine the condenser duty and reboiler heat$-$duty.