Q$4.$ In a plant where the cost of steam is $100TL$ per million $kJ(100T\frac{L}{{10}^6}(kJ)),$ saturated steam at $12$ bar gauge $(\frac{?}{\text{b}\text{a}\text{r}}g)$ is generated while a portion of it is throttled to supply $2$ barg steam. Assuming continuous operation, determine

a $(13$ points$)$: mass flow rate of the low$-$pressure steam produced, and

b $(5$ points$)$: the annual energy savings of producing low$-$pressure steam by flashing $2500k\frac{g}{h}$ of $12$ barg condensate.

c $(7$ points$)$: Draw also $P-h$ diagram for this flashing process.

Data: The average temperature of the boiler makeup water is $20C.$

For $12$ barg: $h_f=814\mathrm{.}59k\frac{J}{k}g$ and $h_g=2786\mathrm{.}5k\frac{J}{k}g$; for $2$ barg: $h_f=561\mathrm{.}43k\frac{J}{k}g,h_g=2724\mathrm{.}9k\frac{J}{k}g.$ Here, the subscripts $f$ and $g$ indicate saturated liquid and saturated vapor, respectively.

For $20C$:$h_f=83\mathrm{.}92k\frac{J}{k}g$; yearly operating duration: $8760h$

Solution: