$($PARTNER PROBLEM$)$ Last semester when doing the project, I mentioned that plants are optimized to use streams from different units to transfer heat $($as needed$).$ In other words, you are trying to maximize your usage of those streams to get the most you can out of the money you spent for those streams $($e$.$g$.,$ those superheated vapor streams used for heating$).$

At any rate, imagine a steam generator $($i$.$e$.,$ a type of boiler$)$ that has multiple feed streams from various parts of a chemical plant, but has one output $($at $300C$ and $\frac{8}{b}ar).$ The inlet $($i$.$e$.,$ feed streams$)$ are actually outlets from three different condensers at different parts of the plant:

$100k\frac{g}{m}in$ of saturated liquid $)=(0$ at $150C$

$200k\frac{g}{m}in$ of saturated liquid $)=(0$ at $180C$

$75k\frac{g}{m}in$ of saturated liquid $)=(0$ at $\frac{6}{b}ar.$

What rate of heat must be added to the steam generator to deliver the superheated steam at the required $T$ and $P?$ Assume the steam generator is operating at steady$-$state. Hint: Write out the $1^{st}$ Law of Thermodynamics and cross out terms, as applicable, to arrive at a final equation you need.